{"id":64,"date":"2021-12-15T09:53:12","date_gmt":"2021-12-15T09:53:12","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/folgen-und-konvergenz\/"},"modified":"2021-12-15T09:53:12","modified_gmt":"2021-12-15T09:53:12","slug":"folgen-und-konvergenz","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/folgen-und-konvergenz\/","title":{"raw":"Folgen und Konvergenz","rendered":"Folgen und Konvergenz"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zf56007f44591\" class=\"sectionHead\"><span class=\"titlemark\">5.3 <\/span> <a id=\"x1-1440003\"><\/a>Folgen und Konvergenz<\/h3> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z783f48e7d787\"> <a id=\"x1-144001r20\"><\/a> <span class=\"ecbx-1095\">Definition 5.20 <\/span>(Folge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge. Eine <span class=\"ecbx-1095\">Folge <\/span>in <math display=\"inline\"><mi>X<\/mi><\/math> ist eine Abbildung <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Das Bild <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> schreibt man auch als <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> und bezeichnet es als das <math display=\"inline\"><mi>n<\/mi><\/math>-te <span class=\"ecbx-1095\">Folgenglied <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Anstatt <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>X<\/mi><\/math> schreibt man auch <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2115<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msubsup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn> <\/mrow> <mrow> <mi>\u221e<\/mi> <\/mrow> <\/msubsup><\/math> oder kurz <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Die Menge der Folgen in <math display=\"inline\"><mi>X<\/mi><\/math> wird auch als <math display=\"inline\"><msup><mrow><mi>X<\/mi><\/mrow><mrow><mi>\u2115<\/mi><\/mrow><\/msup><\/math> bezeichnet. Eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> heisst <span class=\"ecbx-1095\">konstant<\/span>, falls <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> und <span class=\"ecbx-1095\">schliesslich konstant<\/span>, falls ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> existiert mit <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Sei <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>V<\/mi> <\/math> ein Vektorraum \u00fcber <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann bildet die Menge der Folgen in <math display=\"inline\"><mi>V<\/mi> <\/math> zusammen mit den Verkn\u00fcpfungen                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> und Folgen <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>\u2115<\/mi><\/mrow><\/msup><\/math> einen Vektorraum. F\u00fcr <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi><\/math> haben wir dies bereits in den Abschnitten&nbsp;<a href=\"..\/..\/chapter\/reellwertige-funktionen#x1-910004\">3.4<\/a> und <a href=\"..\/..\/chapter\/stetigkeit#x1-950001\">3.5.1<\/a> gesehen (es sind die Vektorr\u00e4ume <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>\u2115<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mi>\u211d<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> respektive <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>\u2115<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mi>\u2102<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>). <a id=\"x1-144002r143\"><\/a> <\/p> <h4 id=\"zde5cdb347488\" class=\"subsectionHead\"><span class=\"titlemark\">5.3.1 <\/span> <a id=\"x1-1450001\"><\/a>Konvergenz von Folgen<\/h4> <p class=\"noindent\">F\u00fcr eine schliesslich konstante Folge&nbsp;<math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in einer Menge <math display=\"inline\"><mi>X<\/mi><\/math> ist&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math> f\u00fcr alle hinreichend grossen&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> eine besondere Zahl, die wir mit der schliesslich konstanten Folge assoziieren k\u00f6nnen. Wir wollen diese Assoziation verallgemeinern, wenn <math display=\"inline\"><mi>X<\/mi><\/math> mit einer Metrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> ausgestattet ist. Dabei erlauben wir eine beliebig kleine Fehlerschranke&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und suchen wiederum ein&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr alle hinreichend grossen&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> das Folgenglied&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> \u2013 bis auf einen Fehler kleiner als&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math> \u2013 gleich&nbsp;<math display=\"inline\"><mi>A<\/mi><\/math> sein soll. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zf205df2242af\"> <a id=\"x1-145001r21\"><\/a> <span class=\"ecbx-1095\">Definition 5.21 <\/span>(Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> ein metrischer Raum und <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math>                                                                                                                                                                           eine Folge in <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> gegen einen Punkt <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecbx-1095\">konvergiert <\/span>oder <span class=\"ecbx-1095\">strebt<\/span>, falls es f\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> gibt, so dass <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> In diesem Fall nennen wir den Punkt <math display=\"inline\"><mi>A<\/mi><\/math> einen <span class=\"ecbx-1095\">Grenzwert <\/span>der Folge und schreiben auch <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> Weiter ist eine Folge in <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecbx-1095\">konvergent<\/span>, falls sie einen Grenzwert besitzt, und <span class=\"ecbx-1095\">divergent<\/span>, falls sie keinen Grenzwert besitzt. <\/p> <\/div> <p class=\"indent\">Nochmals anders (und etwas weniger genau) formuliert ist eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> nach <math display=\"inline\"><mi>A<\/mi><\/math> konvergent, falls hinreichend sp\u00e4te Folgenglieder der Zahl <math display=\"inline\"><mi>A<\/mi><\/math> beliebig nahe kommen. Wir werden uns vorerst haupts\u00e4chlich mit der Untersuchung von Konvergenz in <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math> wie in folgendem Bild besch\u00e4ftigen. Doch wollen wir betonen, dass f\u00fcr die Definition und einige wichtige Eigenschaften der axiomatische Kontext des metrischen Raumes mitunter die Diskussion sogar vereinfachen kann, da diese Diskussion nur auf die Axiome aufbauen kann. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=351&amp;url=Pictures\/folgen\/konvergenzdef.pdf\"><img id=\"z499a0c81b464\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/folgen\/konvergenzdef.svg\" width=\"351\"><\/div>  <\/div> <p class=\"indent\">In Pr\u00e4dikatenlogik ist Konvergenz gegen&nbsp;<math display=\"inline\"><mi>A<\/mi><\/math> durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi> <mo class=\"MathClass-punc\">:<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben. Wir bemerken noch, dass eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in einem metrischen Raum <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> genau dann gegen <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> konvergiert, wenn die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> gegen Null konvergiert. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z65b6c96cc651\"> <a id=\"x1-145002r22\"><\/a> <span class=\"ecbx-1095\">Lemma 5.22.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer          Raum.          Jede          konvergente          Folge          in<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">besitzt einen eindeutigen Grenzwert.<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr eine konvergente Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>X<\/mi><\/math> sprechen wir also von <span class=\"ecbx-1095\">dem <\/span>Grenzwert <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> In Worten l\u00e4sst sich der formale Beweis, den wir gleich geben werden, wie folgt beschreiben. Besitzt eine konvergente Folge (entgegen der Behauptung des Lemmas) zwei verschiedene Grenzwerte, so muss sie sich schlussendlich beliebig nahe an beiden dieser Grenzwerten aufhalten. Nach der Dreiecksungleichung m\u00fcssen diese beiden Grenzwerte also beliebig nahe aneinander liegen, was allerdings nicht m\u00f6glich ist, da sie eine positive Distanz zueinander aufweisen m\u00fcssen. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Seien per Widerspruch <math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> zwei verschiedene Grenzwerte einer konvergenten Folge <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Sei                                                                                                                                                                           <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> gegen <math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> konvergiert, existiert ein <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Genauso existert <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> und <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Nach der Dreiecksungleichung gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was einen Widerspruch darstellt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wir bemerken noch, dass es reicht, die Eigenschaft in der Definition der Konvergenz f\u00fcr <span class=\"ecti-1095\">kleine<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> zu pr\u00fcfen \u2013 siehe folgende \u00dcbung. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zbd1adf6f9c79\"> <a id=\"x1-145003r23\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.23.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge in einem metrischen Raum <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und sei <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">genau dann gegen <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">konvergiert, wenn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">existiert mit <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Konvergenz l\u00e4sst sich bequem mit offenen B\u00e4llen oder sogenannten Umgebungen beschreiben. Wir erinnern daran, dass f\u00fcr einen metrischen <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> der <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-Ball oder auch die <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math><span class=\"ecbx-1095\">-Umgebung<\/span> um <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben ist (siehe Definition&nbsp;<a href=\"..\/..\/chapter\/metrische-raeume#x1-142001r16\">5.16<\/a>). Eine allgemeine Umgebung ist wie folgt definiert. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zf030ce3c4fde\"> <a id=\"x1-145004r24\"><\/a> <span class=\"ecbx-1095\">Definition 5.24 <\/span>(Umgebungen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> ein metrischer Raum. Eine <span class=\"ecbx-1095\">Umgebung <\/span>von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> ist eine Teilmenge <span class=\"maperiod\"><math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> die eine <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-Umgebung                                                                                                                                                                           von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr ein <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> enth\u00e4lt. <\/p> <\/div> <p class=\"indent\">Die obige Definition von Umgebungen erlaubt nun eine alternative Formulierung von Konvergenz: Eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in einem metrischen Raum <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert genau dann gegen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> wenn f\u00fcr jede Umgebung <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecbx-1095\">fast alle <\/span>(das heisst, alle bis auf endlich viele) Folgenglieder von <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in <math display=\"inline\"><mi>V<\/mi> <\/math> liegen (wieso?). <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z5da1fc5abf69\"> <a id=\"x1-145005r25\"><\/a> <span class=\"ecbx-1095\">Lemma 5.25 <\/span>(Indexverschiebung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in einem<\/span> <span class=\"ecti-1095\">metrischen Raum und <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">genau dann konvergent<\/span> <span class=\"ecti-1095\">wenn die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent ist. In diesem Fall gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zde82f530306b\"> <a id=\"x1-145006r26\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.26.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Lemma <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145005r25\"><span class=\"ecti-1095\">5.25<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <p class=\"indent\">Da nach Lemma&nbsp;<a href=\"..\/..\/chapter\/metrische-raeume#x1-140002r11\">5.11<\/a> jeder normierte Vektorraum <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>V<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> eine Metrik induziert, erhalten wir einen Konvergenzbegriff f\u00fcr Folgen in <span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> Explizit ausgedr\u00fcckt konvergiert dann eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>V<\/mi> <\/math> gegen <span class=\"maperiod\"><math display=\"inline\"><mstyle><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub><\/mstyle><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math><\/span><span class=\"period\">,<\/span> wenn f\u00fcr alle <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> existiert, so dass f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in einem normierten Vektorraum <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>V<\/mi><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> heisst <span class=\"ecbx-1095\">beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt<\/span>, falls es ein <math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt, so dass <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wie in \u00dcbung <a href=\"..\/..\/chapter\/reellwertige-funktionen#x1-92002r38\">3.38<\/a> kann man zeigen, dass die Menge der beschr\u00e4nkten Folgen in <math display=\"inline\"><mi>V<\/mi> <\/math> einen Unterraum des Vektorraums der Folgen in <math display=\"inline\"><mi>V<\/mi> <\/math> bildet.                                                                                                                                                                           <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"ze32f870511be\"> <a id=\"x1-145007r27\"><\/a> <span class=\"ecbx-1095\">Lemma 5.27 <\/span>(Beschr\u00e4nktheit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Jede konvergente        Folge        in        einem        normierten        Vektorraum<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>V<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> eine konvergente Folge und <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Dann existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daraus folgt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-145008r145\"><\/a> <h4 id=\"z248e123a23ae\" class=\"subsectionHead\"><span class=\"titlemark\">5.3.2 <\/span> <a id=\"x1-1460002\"><\/a>Erste Konsequenzen und Beispiele<\/h4> <p class=\"noindent\">Wir empfehlen den Leserinnen und Lesern sich in diesem Unterabschnitt auf Folgen in <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math> zu konzentrieren. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zac77def40604\"> <a id=\"x1-146001r28\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.28 <\/span>(Konvergente und divergente Folgen in <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Eine konstante Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">konvergiert gegen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Genauso konvergieren schliesslich konstante Folgen gegen den Wert, den sie schliesslich<\/span> <span class=\"ecti-1095\">annehmen.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert gegen Null, das heisst <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Denn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert nach dem Archimedischen Prinzip (Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\"><span class=\"ecti-1095\">2.68<\/span><\/a><span class=\"ecti-1095\">) ein <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>N<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">gilt nun <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>N<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">(und damit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">).<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegeben durch <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ist divergent, da die Folgenglieder <\/span><math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/math> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">hin und her wechseln und sich insbesondere keiner bestimmten Zahl n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hern.<\/span> <p class=\"noindent\"><span class=\"ecti-1095\">Formal argumentiert: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Zahl <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist entweder <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und somit <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">oder <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Im ersten Fall gibt es f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein gerades <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und damit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">(anstatt <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">).<\/span> <span class=\"ecti-1095\">Im zweiten Fall gibt es f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein ungerades <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">(anstatt <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><\/math><span class=\"ecti-1095\">).<\/span><\/p><\/div><\/div> <\/div> <p class=\"indent\">Nach obigem Beispiel k\u00f6nnte man sich die Frage stellen, ob das Konvergenzverhalten einer Folge reeller Zahlen in <math display=\"inline\"><mi>\u2102<\/mi><\/math> dasselbe ist, wenn man die Folge als Folge in <math display=\"inline\"><mi>\u211d<\/mi><\/math> betrachtet. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z6796bd396be6\"> <a id=\"x1-146002r29\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.29 <\/span>(Reelle Grenzwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente Folge in <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">reell ist.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Nehmen                          Sie                          an,                          dass<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und                                                w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen                                                Sie<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">so,                 dass                 der                 Ball                 von                 Radius<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">um<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">die reelle Zahlengerade nicht schneidet.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wie schon bei der Stetigkeit von Funktionen m\u00f6chten wir auch hier nicht jedesmal \u201evon Hand\u201c mit&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und&nbsp;<math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math> Grenzwerte berechnen m\u00fcssen. Dazu ist folgende Proposition hilfreich. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z7552e1a0c190\"> <a id=\"x1-146003r30\"><\/a> <span class=\"ecbx-1095\">Proposition 5.30 <\/span>(Additive und multiplikative Eigenschaften des Grenzwerts)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">zwei konvergente<\/span> <span class=\"ecti-1095\">Folgen in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist konvergent und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist konvergent und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die Folge <\/span><math display=\"inline\"><mi>\u03b1<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <span class=\"ecti-1095\">die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere bildet die Menge der konvergenten Folgen in<\/span> <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>\u2115<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">einen<\/span> <span class=\"ecti-1095\">Unterraum und der Grenzwert stellt eine lineare Abbildung von diesem Unterraum nach<\/span> <math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">dar.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir setzen <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">F\u00fcr (i) sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> und <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Sei <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach der Dreiecksungleichung ist f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was die Aussage in (i) impliziert. <\/p><p class=\"indent\">F\u00fcr (ii) bemerken wir zuerst, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und m\u00f6chten die letzteren beiden Terme einzeln absch\u00e4tzen. Dabei m\u00fcssen wir sicher stellen, dass <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr grosse <math display=\"inline\"><mi>n<\/mi><\/math> nicht zu gross wird (siehe auch Lemma&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145007r27\">5.27<\/a>). Sei <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> (\u00e4hnlich wie in (i)) so gew\u00e4hlt, dass f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> }<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dann gilt insbesondere <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Damit ist f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was nach obiger Absch\u00e4tzung f\u00fcr <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> die Aussage in (ii) beweist. <\/p><p class=\"indent\">Die Behauptung in (i) und (ii) implizieren auch die letzte Aussage in der Proposition, womit nur noch (iii) zu beweisen ist. Also angenommen <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>A<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir sehen also, dass wir erzwingen k\u00f6nnen, dass <math display=\"inline\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>A<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/math> klein ist, wenn <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> klein ist. Dazu m\u00fcssen wir allerdings verhindern, dass                                                                                                                                                                           <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> zu klein wird. F\u00fcr den formalen Beweis sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Nach Definition von <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> gilt dann nach der umgekehrten Dreiecksungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Also wird <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nicht zu klein und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>A<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2215<\/mo><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2215<\/mo><mn>2<\/mn><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2215<\/mo><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was zu zeigen war. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za0e7c6506469\"> <a id=\"x1-146007r31\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.31.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Vergleichen Sie die Argumente f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Proposition <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94008r50\"><span class=\"ecti-1095\">3.50<\/span><\/a> <span class=\"ecti-1095\">mit dem Beweis von (i) und (ii) in<\/span> <span class=\"ecti-1095\">Proposition <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\"><span class=\"ecti-1095\">5.30<\/span><\/a><span class=\"ecti-1095\">. Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren Sie auch, welche der bewiesenen Aussagen auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r normierte<\/span> <span class=\"ecti-1095\">Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume gelten und wieso.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z6b3dd61cd42d\"> <a id=\"x1-146008r32\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.32 <\/span>(Rationale Funktionen als Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie folgende Grenzwerte, wenn sie existieren:<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mn>7<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mn>5<\/mn><\/mrow> <mrow><mn>3<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>0<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Formulieren und beweisen Sie allgemeine Versionen von den Beispielen in (i).<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Verwenden Sie hier und auch sonst kein fr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">her erlerntes Kochrezept, das Sie nicht begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nden<\/span> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Stattdessen erweitern Sie mit <\/span><math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein geeignetes <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und argumentieren Sie unter Verwendung des obigen Wissens (Beispiel <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146001r28\"><span class=\"ecti-1095\">5.28<\/span><\/a> <span class=\"ecti-1095\">und Proposition<\/span> <a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\"><span class=\"ecti-1095\">5.30<\/span><\/a><span class=\"ecti-1095\">).<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Eine konvergente Folge in <math display=\"inline\"><mi>\u2102<\/mi><\/math> (oder allgemeiner in einem normierten Vektorraum) mit Grenzwert Null wird auch eine <span class=\"ecbx-1095\">Nullfolge<\/span> genannt. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7cd4c5c0d5ec\"> <a id=\"x1-146011r33\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.33 <\/span>(Nullfolgen und Divergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine komplex-wertige Folge mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegen <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">konvergiert. Zeigen Sie, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">divergiert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5e70c81ba8c7\"> <a id=\"x1-146012r34\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.34 <\/span>(Geometrische Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bezeichnen wir als <\/span><span class=\"ecbi-1095\">geometrische Folge <\/span><span class=\"ecti-1095\">zum Skalierungsfaktor<\/span> <math display=\"inline\"><mi>q<\/mi><\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">untersuchen nun diese geometrische Folge auf Konvergenz.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">wissen wir bereits, dass die Folge <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">divergiert (also keinen Grenzwert hat).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Allgemeiner gilt, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und divergiert ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und daher divergent.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen noch (iii)-(v) beweisen. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii) argumentieren wir indirekt. Angenommen<\/span> <math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>A<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">nach Proposition <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\"><span class=\"ecti-1095\">5.30<\/span><\/a> <span class=\"ecti-1095\">und Lemma <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145005r25\"><span class=\"ecti-1095\">5.25<\/span><\/a><span class=\"ecti-1095\">. Dies impliziert<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und wegen<\/span> <math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">, dass<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Da aber<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gilt,<\/span> <span class=\"ecti-1095\">kann <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">nicht der Grenzwert der Folge sein.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iv) sei nun <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">. Wir zeigen, dass die Folge<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist, womit (iv)<\/span> <span class=\"ecti-1095\">aus Lemma <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145007r27\"><span class=\"ecti-1095\">5.27<\/span><\/a> <span class=\"ecti-1095\">folgt. Sei <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nach dem Archimedischen Prinzip (Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\"><span class=\"ecti-1095\">2.68<\/span><\/a><span class=\"ecti-1095\">) existiert ein<\/span> <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>N<\/mi><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>M<\/mi><\/math><span class=\"ecti-1095\">. Nun ergibt die<\/span> <span class=\"ecti-1095\">Bernoulli-Ungleichung (Lemma <\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-79001r5\"><span class=\"ecti-1095\">3.5<\/span><\/a><span class=\"ecti-1095\">) <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>N<\/mi><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">womit die Behauptung gezeigt ist.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (v) sei <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Falls<\/span> <math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">so ist<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Sei nun<\/span> <math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Da<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">, existiert wegen<\/span> <span class=\"ecti-1095\">(iv) ein <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">, so<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>N<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\ud835\udf00<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Somit<\/span> <span class=\"ecti-1095\">gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z39d274814479\"> <a id=\"x1-146018r35\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.35.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>n<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Zeigen Sie zuerst mit Hilfe der Bernoulli-Ungleichung, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z47fddf9b4b99\"> <a id=\"x1-146019r36\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.36 <\/span>(Ces\u00e0ro-Mittel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">konvergente Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Folge der <\/span><span class=\"ecbi-1095\">Ces<\/span><span class=\"ecbi-1095\">\u00e0<\/span><span class=\"ecbi-1095\">ro-Mittel <\/span><span class=\"ecti-1095\">(auch <\/span><span class=\"ecbi-1095\">arithmetische Mittel <\/span><span class=\"ecti-1095\">oder <\/span><span class=\"ecbi-1095\">Cauchy-Mittel<\/span> <span class=\"ecti-1095\">genannt) <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegeben durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">konvergiert und<\/span> <span class=\"ecti-1095\">denselben Grenzwert wie <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">hat.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berzeugen Sie sich auch davon, dass die umgekehrte Implikation nicht gilt, das heisst, dass die<\/span> <span class=\"ecti-1095\">Konvergenz der Ces<\/span><span class=\"ecti-1095\">\u00e0<\/span><span class=\"ecti-1095\">ro-Mittel nicht Konvergenz der Folge impliziert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z94b0969d646d\"> <a id=\"x1-146020r37\"><\/a> <span class=\"ecbx-1095\">Applet 5.37 <\/span>(Einige Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"391px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/FgDxWNhj\/width\/688\/height\/391\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten verschiedene Folgen und k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen mittels Verkleinern der <\/span><math display=\"inline\"><mi>x<\/mi><\/math><span class=\"ecti-1095\">-Achse<\/span> <span class=\"ecti-1095\">die Konvergenz- und Divergenzeigenschaften der Folgen beobachten.<\/span> <\/p> <\/div> <a id=\"x1-146021r146\"><\/a> <h4 id=\"z7dae186e48c0\" class=\"subsectionHead\"><span class=\"titlemark\">5.3.3 <\/span> <a id=\"x1-1470003\"><\/a>Teilfolgen<\/h4> <p class=\"noindent\">Oft m\u00f6chte man anstelle einer Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nur einen \u201e Teil\u201c der Folge betrachten, wobei wir im Gegensatz zur Indexverschiebung in Lemma&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145005r25\">5.25<\/a> manchmal auch unendlich viele Folgenglieder wegstreichen wollen. Dies ist insbesondere dann der Fall, wenn die Folge nicht konvergiert. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zd2983185910b\"> <a id=\"x1-147001r38\"><\/a> <span class=\"ecbx-1095\">Definition 5.38 <\/span>(Teilfolge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Wenn <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> eine                       Folge                       in                       einer                       Menge <math display=\"inline\"><mi>X<\/mi><\/math> ist                                                                                                                  und <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> eine streng         monoton         wachsende         Folge         ist,         dann         wird <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> eine                                                  <span class=\"ecbx-1095\">Teilfolge                                           <\/span>von <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> genannt. <\/p> <\/div> <p class=\"indent\">Liegt eine Teilfolge einer konvergenten Folge vor, so konvergiert diese gegen denselben Grenzwert wie die Folge. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"zac3a5a4dfdef\"> <a id=\"x1-147002r39\"><\/a> <span class=\"ecbx-1095\">Lemma 5.39 <\/span>(Konvergenz von Teilfolgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente Folge in einem metrischen Raum <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Jede Teilfolge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">konvergiert und hat denselben Grenzwert <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z8dfc21f50b29\"> <a id=\"x1-147003r40\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.40.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Lemma <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-147002r39\"><span class=\"ecti-1095\">5.39<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r          eine          streng          monoton          wachsende          Folge<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>k<\/mi><\/math> <span class=\"ecti-1095\">(wieso?).<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Eine Folge kann konvergente Teilfolgen besitzen, ohne selbst zu konvergieren. Beispielsweise hat die Folge <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die konvergente (konstante) Teilfolge <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> konvergiert aber nicht, wie wir schon gesehen haben. In der Tat haben wir mit Lemma&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-147002r39\">5.39<\/a> jetzt ein k\u00fcrzeres Argument. Falls die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> gegen <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> konvergieren w\u00fcrde, so m\u00fcssten die beiden konstanten Folgen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> auch gegen <math display=\"inline\"><mi>A<\/mi><\/math> konvergieren. Dies ist nat\u00fcrlich nicht m\u00f6glich, da die eine gegen                                                                                                                                                                           <math display=\"inline\"><mn>1<\/mn><\/math> und die andere gegen <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> konvergiert. <\/p><p class=\"indent\">In gewissen Situationen l\u00e4sst sich aus dem Konvergenzverhalten von Teilfolgen trotzdem etwas \u00fcber das Konvergenzverhalten der gesamten Folge sagen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z74a7f2112239\"> <a id=\"x1-147004r41\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.41 <\/span>(Teilfolgen von Teilfolgen und Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge in <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">genau dann gegen <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">konvergiert, wenn jede Teilfolge von <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Teilfolge besitzt, die gegen <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Betrachten                                        Sie                                        zu<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die                                                                                                      Menge<\/span> <math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo> <mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>\ud835\udf00<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und zeigen Sie indirekt, dass diese endlich sein muss.<\/span><\/p><\/details>  <\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z9c09931fda6c\"> <a id=\"x1-147005r42\"><\/a> <span class=\"ecbx-1095\">Proposition 5.42 <\/span>(H\u00e4ufungspunkte einer Folge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge in<\/span> <span class=\"ecti-1095\">einem metrischen Raum <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Ein Punkt <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">heisst<\/span> <span class=\"ecbi-1095\">H<\/span><span class=\"ecbi-1095\">\u00e4<\/span><span class=\"ecbi-1095\">ufungspunkt <\/span><span class=\"ecti-1095\">von <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">falls die folgenden <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalenten Bedingungen erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt sind.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gibt eine Teilfolge <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gibt es ein <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen (a) gilt. Sei also <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> eine konvergente Teilfolge von <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> mit Grenzwert <math display=\"inline\"><mi>A<\/mi><\/math> und sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert ein <math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>K<\/mi><\/math><\/span><span class=\"period\">.<\/span> Sei nun <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>K<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann erf\u00fcllt <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> die Bedingung <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> wie gewollt und (b) ist erf\u00fcllt. <\/p><p class=\"indent\">Angenommen (b) gilt. Wir m\u00f6chten rekursiv eine Teilfolge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> finden mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Diese konvergiert dann gegen <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> da f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> die Ungleichung <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math>                                                                                                                                                                           f\u00fcr alle <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\ud835\udf00<\/mi><\/mrow><\/mfrac><\/math> erf\u00fcllt ist. <\/p><p class=\"indent\">Sei <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gibt es ein <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Nun nehmen wir an, dass <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> bereits konstruiert sind mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u2113<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>k<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir setzen <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert nach Voraussetzung ein <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies beendet den Induktionsschritt und wir erhalten durch Rekursion die gew\u00fcnschte Teilfolge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> mit Grenzwert <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z13973e47d1cc\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Obiger                                                                                                    Beweis ist formal nicht ganz unproblematisch. Denn zum Unterschied von der Rekursion, welche wir am Ende von Abschnitt <a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-510001\">2.2.1<\/a> besprochen haben, m\u00fcssen wir hier eigentlich eine Wahl f\u00fcr <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> treffen. Da diese Wahl nicht nur einmal notwendig ist, sondern abz\u00e4hlbar oft, so haben wir in obiger Formulierung eigentlich eine (schwache) Version des Auswahlaxioms verwendet. Wir werden uns diese Freiheit hier und auch in \u00e4hnlichen Situationen erlauben, ohne dieses Auswahlaxiom  genauer  zu  besprechen.  Mit  ein  Grund  daf\u00fcr  ist,  dass  ein  Grossteil  der modernen Mathematik an diesem Auswahlaxiom der Axiomatischen Mengenlehre gebunden ist. Es ist aber auch m\u00f6glich (wenn auch anstrengend) die Verwendung dieses Auswahlaxioms in obigem Beweis zu vermeiden indem man bei jeder Wahl sicherstellt, dass man das minimale <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit allen gew\u00fcnschten Eigenschaften verwendet. <\/p> <\/div> <a id=\"x1-147008r147\"><\/a> <h4 id=\"z899045bf5002\" class=\"subsectionHead\"><span class=\"titlemark\">5.3.4 <\/span> <a id=\"x1-1480004\"><\/a>Konvergenz in endlich-dimensionalen Vektorr\u00e4umen<\/h4> <p class=\"noindent\">Im Allgemeinen h\u00e4ngt der Konvergenzbegriff auf einer Menge <math display=\"inline\"><mi>X<\/mi><\/math> von der Metrik ab, die man auf <math display=\"inline\"><mi>X<\/mi><\/math> betrachtet. Folgende \u00dcbung enth\u00e4lt ein Beispiel. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf7f1abe21ca1\"> <a id=\"x1-148001r43\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.43 <\/span>(Manhattan und SNCF sind sehr verschieden)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Finden Sie eine Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die zwar bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Manhattanmetrik, aber nicht bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der franz<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sischen Eisenbahnmetrik<\/span> <span class=\"ecti-1095\">konvergiert (wobei wir <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und damit <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit einer Teilmenge von <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">identifizieren).<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr normierte endlich-dimensionale Vektorr\u00e4ume ist die Situation oft vorteilshafter. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z31334a5bac0a\"> <a id=\"x1-148002r44\"><\/a> <span class=\"ecbx-1095\">Proposition 5.44.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">, sei<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi> <\/mrow>  <\/math> <span class=\"ecti-1095\">eine Folge<\/span> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">, und<\/span> <span class=\"ecti-1095\">sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Folgende Aussagen sind <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> <span class=\"ecti-1095\">konvergiert gegen <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> <span class=\"ecti-1095\">konvergiert gegen <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> <span class=\"ecti-1095\">konvergiert gegen <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> <span class=\"ecti-1095\">konvergiert die Folge der Komponenten <\/span><math display=\"inline\"><msub><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/msub><mo fence=\"true\" form=\"postfix\">)<\/mo><mo fence=\"true\" form=\"postfix\">)<\/mo> <mrow><mi>n<\/mi><\/mrow><\/math> <span class=\"ecti-1095\">gegen <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Inbesondere gilt diese <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenz auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Folge in<\/span> <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">In der Tat werden wir sp\u00e4ter sehen, dass man in obiger Proposition eine beliebige Norm auf <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> betrachen kann. Auf Grund von Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a> werden wir oft von Konvergenz einer Folge in <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> oder                                                                                                                                                                           <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> sprechen, ohne die Norm anzugeben. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir beweisen zuerst die \u00c4quivalenz der Aussagen in (i), (ii), und (iii) und verwenden daf\u00fcr die Ungleichungen <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>d<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;und<\/mtext><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-148007r2\" \/><mstyle class=\"maketag\"><mtext>(5.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msqrt><mrow><mi>d<\/mi><\/mrow><\/msqrt><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-148008r3\" \/><mstyle class=\"maketag\"><mtext>(5.3)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> In der Tat behauptet die erste Ungleichung (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148007r2\">5.2<\/a>) bloss, dass der maximale Absolutbetrag kleiner gleich der Summe der Absolutbetr\u00e4ge, und die Summe der Absolutbetr\u00e4ge kleiner gleich <math display=\"inline\"><mi>d<\/mi><\/math> mal dem maximalen Absolutbetrages ist. Die Ungleichung (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148008r3\">5.3<\/a>) ergibt sich analog aus <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msubsup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msubsup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>d<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msubsup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mi>\u221e<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span>                                                                                                                                                                           <\/p><p class=\"indent\">Unter Verwendung der Ungleichungen in (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148007r2\">5.2<\/a>) und (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148008r3\">5.3<\/a>) ist der Beweis der \u00c4quivalenz der Aussagen in (i), (ii) und (iii) ziemlich direkt. Zur Illustration beweisen wir (i) <math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><\/math> (ii); alle anderen Implikationen verfiziert man analog. Sei also <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und sei <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi> <\/mrow>  <mo class=\"MathClass-bin\">\u2212<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mi>d<\/mi><\/mrow><\/mfrac><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> (wir verwenden hier, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi> <\/mrow>  <\/math> nach Annahme bez\u00fcglich der Norm <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> gegen <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math> konvergiert). Nach (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148007r2\">5.2<\/a>) gilt f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>d<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was (i) <math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <\/math> (ii) beweist. <\/p><p class=\"indent\">Zum Schluss beweisen wir nun die \u00c4quivalenz der Aussagen in (i) und (iv). Angenommen (i) gilt. Somit gibt es f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi> <\/mrow>  <mo class=\"MathClass-bin\">\u2212<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere gilt f\u00fcr <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">,<\/span> womit <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> gegen <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert, da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war. <\/p><p class=\"indent\">Wir nehmen nun umgekehrt (iv) an, also dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi> <\/mrow>  <\/math> gegen <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert f\u00fcr jedes <span class=\"maperiod\"><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gibt es zu <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> ein <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi> <\/mrow>  <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Sei <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dann gilt f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig ist, folgt die Konvergenz von <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> gegen <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math><\/span><span class=\"period\">,<\/span> was das Lemma beweist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zb59352108958\"> <a id=\"x1-148009r45\"><\/a> <span class=\"ecbx-1095\">Korollar 5.45 <\/span>(Reduktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine komplexwertige Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist genau dann konvergent (mit Grenzwert <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><span class=\"ecti-1095\">),<\/span> <span class=\"ecti-1095\">wenn die beiden reellwertigen Folgen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent sind (mit Grenzwerten <\/span><math display=\"inline\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir identifizieren <math display=\"inline\"><mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> (gewissermassen tautologisch) via der Bijektion <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dann ist <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> womit eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>\u2102<\/mi><\/math> genau dann gegen <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> konvergiert, wenn <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> gegen <math display=\"inline\"><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert. Hiermit folgt das Korollar in der Tat aus Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a>(iv). <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2d7d2a07a5c2\"> <a id=\"x1-148010r46\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.46.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert und geben Sie den Grenzwert an. Impliziert umgekehrt die Konvergenz von <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">die Konvergenz von <\/span><span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"endquote\">?<\/span> <\/p> <\/div> <p class=\"indent\">Folgende \u00dcbung gibt ein Beispiel eines unendlich-dimensionalen Vektorraums <math display=\"inline\"><mi>V<\/mi> <\/math> und zweier Normen auf <span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">,<\/span> die einen unterschiedlichen Konvergenzbegriff definieren. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9a6667fbc069\"> <a id=\"x1-148011r47\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.47 <\/span>(<math display=\"inline\"><mn>1<\/mn><\/math>-Norm und Konvergenz im Mittel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ein kompaktes<\/span> <span class=\"ecti-1095\">Intervall mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">. Wir betrachten<\/span> <span class=\"ecti-1095\">den Vektorraum <\/span><math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>C<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit den Normen <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">definiert in Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-1380002\"><span class=\"ecti-1095\">5.1.2<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass Konvergenz <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <span class=\"ecti-1095\">auch die Konvergenz <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">impliziert.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie eine Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>V<\/mi> <\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn> <\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Konvergenz bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">nennt man auch gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssige Konvergenz; wir werden diese sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter in diesem Semester<\/span> <span class=\"ecti-1095\">nochmals einf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren und genauer untersuchen. Konvergenz bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">nennt<\/span> <span class=\"ecti-1095\">man auch Konvergenz im Mittel. Diese <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung hat also gezeigt, dass Konvergenz im Mittel und<\/span> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssige Konvergenz verschiedene Begriffe sind.<\/span> <\/p> <\/div> <a id=\"x1-148014r144\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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margin-top:0.25em; margin-left:1em; }\n.abstract p {margin-left:5%; margin-right:5%;}\ndiv.abstract {width:100%;}\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zf56007f44591\" class=\"sectionHead\"><span class=\"titlemark\">5.3 <\/span> <a id=\"x1-1440003\"><\/a>Folgen und Konvergenz<\/h3> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z783f48e7d787\"> <a id=\"x1-144001r20\"><\/a> <span class=\"ecbx-1095\">Definition 5.20 <\/span>(Folge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge. Eine <span class=\"ecbx-1095\">Folge <\/span>in <math display=\"inline\"><mi>X<\/mi><\/math> ist eine Abbildung <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Das Bild <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> schreibt man auch als <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> und bezeichnet es als das <math display=\"inline\"><mi>n<\/mi><\/math>-te <span class=\"ecbx-1095\">Folgenglied <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Anstatt <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>X<\/mi><\/math> schreibt man auch <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2115<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msubsup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn> <\/mrow> <mrow> <mi>\u221e<\/mi> <\/mrow> <\/msubsup><\/math> oder kurz <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Die Menge der Folgen in <math display=\"inline\"><mi>X<\/mi><\/math> wird auch als <math display=\"inline\"><msup><mrow><mi>X<\/mi><\/mrow><mrow><mi>\u2115<\/mi><\/mrow><\/msup><\/math> bezeichnet. Eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> heisst <span class=\"ecbx-1095\">konstant<\/span>, falls <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> und <span class=\"ecbx-1095\">schliesslich konstant<\/span>, falls ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> existiert mit <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Sei <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>V<\/mi> <\/math> ein Vektorraum \u00fcber <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann bildet die Menge der Folgen in <math display=\"inline\"><mi>V<\/mi> <\/math> zusammen mit den Verkn\u00fcpfungen                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> und Folgen <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>\u2115<\/mi><\/mrow><\/msup><\/math> einen Vektorraum. F\u00fcr <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi><\/math> haben wir dies bereits in den Abschnitten&nbsp;<a href=\"..\/..\/chapter\/reellwertige-funktionen#x1-910004\">3.4<\/a> und <a href=\"..\/..\/chapter\/stetigkeit#x1-950001\">3.5.1<\/a> gesehen (es sind die Vektorr\u00e4ume <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>\u2115<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mi>\u211d<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> respektive <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>\u2115<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>F<\/mi><\/mrow><mrow><mi>\u2102<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>). <a id=\"x1-144002r143\"><\/a> <\/p> <h4 id=\"zde5cdb347488\" class=\"subsectionHead\"><span class=\"titlemark\">5.3.1 <\/span> <a id=\"x1-1450001\"><\/a>Konvergenz von Folgen<\/h4> <p class=\"noindent\">F\u00fcr eine schliesslich konstante Folge&nbsp;<math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in einer Menge <math display=\"inline\"><mi>X<\/mi><\/math> ist&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math> f\u00fcr alle hinreichend grossen&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> eine besondere Zahl, die wir mit der schliesslich konstanten Folge assoziieren k\u00f6nnen. Wir wollen diese Assoziation verallgemeinern, wenn <math display=\"inline\"><mi>X<\/mi><\/math> mit einer Metrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> ausgestattet ist. Dabei erlauben wir eine beliebig kleine Fehlerschranke&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und suchen wiederum ein&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr alle hinreichend grossen&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> das Folgenglied&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> \u2013 bis auf einen Fehler kleiner als&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math> \u2013 gleich&nbsp;<math display=\"inline\"><mi>A<\/mi><\/math> sein soll. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf205df2242af\"> <a id=\"x1-145001r21\"><\/a> <span class=\"ecbx-1095\">Definition 5.21 <\/span>(Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> ein metrischer Raum und <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math>                                                                                                                                                                           eine Folge in <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> gegen einen Punkt <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecbx-1095\">konvergiert <\/span>oder <span class=\"ecbx-1095\">strebt<\/span>, falls es f\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> gibt, so dass <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> In diesem Fall nennen wir den Punkt <math display=\"inline\"><mi>A<\/mi><\/math> einen <span class=\"ecbx-1095\">Grenzwert <\/span>der Folge und schreiben auch <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> Weiter ist eine Folge in <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecbx-1095\">konvergent<\/span>, falls sie einen Grenzwert besitzt, und <span class=\"ecbx-1095\">divergent<\/span>, falls sie keinen Grenzwert besitzt. <\/p> <\/div> <p class=\"indent\">Nochmals anders (und etwas weniger genau) formuliert ist eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> nach <math display=\"inline\"><mi>A<\/mi><\/math> konvergent, falls hinreichend sp\u00e4te Folgenglieder der Zahl <math display=\"inline\"><mi>A<\/mi><\/math> beliebig nahe kommen. Wir werden uns vorerst haupts\u00e4chlich mit der Untersuchung von Konvergenz in <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math> wie in folgendem Bild besch\u00e4ftigen. Doch wollen wir betonen, dass f\u00fcr die Definition und einige wichtige Eigenschaften der axiomatische Kontext des metrischen Raumes mitunter die Diskussion sogar vereinfachen kann, da diese Diskussion nur auf die Axiome aufbauen kann. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=351&amp;url=Pictures\/folgen\/konvergenzdef.pdf\"><img decoding=\"async\" id=\"z499a0c81b464\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/folgen\/konvergenzdef.svg\" width=\"351\" \/><\/div>  <\/div> <p class=\"indent\">In Pr\u00e4dikatenlogik ist Konvergenz gegen&nbsp;<math display=\"inline\"><mi>A<\/mi><\/math> durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi> <mo class=\"MathClass-punc\">:<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben. Wir bemerken noch, dass eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in einem metrischen Raum <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> genau dann gegen <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> konvergiert, wenn die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> gegen Null konvergiert. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z65b6c96cc651\"> <a id=\"x1-145002r22\"><\/a> <span class=\"ecbx-1095\">Lemma 5.22.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer          Raum.          Jede          konvergente          Folge          in<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">besitzt einen eindeutigen Grenzwert.<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr eine konvergente Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>X<\/mi><\/math> sprechen wir also von <span class=\"ecbx-1095\">dem <\/span>Grenzwert <span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> In Worten l\u00e4sst sich der formale Beweis, den wir gleich geben werden, wie folgt beschreiben. Besitzt eine konvergente Folge (entgegen der Behauptung des Lemmas) zwei verschiedene Grenzwerte, so muss sie sich schlussendlich beliebig nahe an beiden dieser Grenzwerten aufhalten. Nach der Dreiecksungleichung m\u00fcssen diese beiden Grenzwerte also beliebig nahe aneinander liegen, was allerdings nicht m\u00f6glich ist, da sie eine positive Distanz zueinander aufweisen m\u00fcssen. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Seien per Widerspruch <math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> zwei verschiedene Grenzwerte einer konvergenten Folge <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Sei                                                                                                                                                                           <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> gegen <math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> konvergiert, existiert ein <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Genauso existert <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> und <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Nach der Dreiecksungleichung gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was einen Widerspruch darstellt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wir bemerken noch, dass es reicht, die Eigenschaft in der Definition der Konvergenz f\u00fcr <span class=\"ecti-1095\">kleine<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> zu pr\u00fcfen \u2013 siehe folgende \u00dcbung. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zbd1adf6f9c79\"> <a id=\"x1-145003r23\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.23.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge in einem metrischen Raum <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und sei <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">genau dann gegen <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">konvergiert, wenn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">existiert mit <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Konvergenz l\u00e4sst sich bequem mit offenen B\u00e4llen oder sogenannten Umgebungen beschreiben. Wir erinnern daran, dass f\u00fcr einen metrischen <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> der <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-Ball oder auch die <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math><span class=\"ecbx-1095\">-Umgebung<\/span> um <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben ist (siehe Definition&nbsp;<a href=\"..\/..\/chapter\/metrische-raeume#x1-142001r16\">5.16<\/a>). Eine allgemeine Umgebung ist wie folgt definiert. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf030ce3c4fde\"> <a id=\"x1-145004r24\"><\/a> <span class=\"ecbx-1095\">Definition 5.24 <\/span>(Umgebungen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> ein metrischer Raum. Eine <span class=\"ecbx-1095\">Umgebung <\/span>von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> ist eine Teilmenge <span class=\"maperiod\"><math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> die eine <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-Umgebung                                                                                                                                                                           von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr ein <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> enth\u00e4lt. <\/p> <\/div> <p class=\"indent\">Die obige Definition von Umgebungen erlaubt nun eine alternative Formulierung von Konvergenz: Eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in einem metrischen Raum <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert genau dann gegen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> wenn f\u00fcr jede Umgebung <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecbx-1095\">fast alle <\/span>(das heisst, alle bis auf endlich viele) Folgenglieder von <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in <math display=\"inline\"><mi>V<\/mi> <\/math> liegen (wieso?). <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5da1fc5abf69\"> <a id=\"x1-145005r25\"><\/a> <span class=\"ecbx-1095\">Lemma 5.25 <\/span>(Indexverschiebung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in einem<\/span> <span class=\"ecti-1095\">metrischen Raum und <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">genau dann konvergent<\/span> <span class=\"ecti-1095\">wenn die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent ist. In diesem Fall gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zde82f530306b\"> <a id=\"x1-145006r26\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.26.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Lemma <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145005r25\"><span class=\"ecti-1095\">5.25<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <p class=\"indent\">Da nach Lemma&nbsp;<a href=\"..\/..\/chapter\/metrische-raeume#x1-140002r11\">5.11<\/a> jeder normierte Vektorraum <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>V<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> eine Metrik induziert, erhalten wir einen Konvergenzbegriff f\u00fcr Folgen in <span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> Explizit ausgedr\u00fcckt konvergiert dann eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>V<\/mi> <\/math> gegen <span class=\"maperiod\"><math display=\"inline\"><mstyle><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub><\/mstyle><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math><\/span><span class=\"period\">,<\/span> wenn f\u00fcr alle <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> existiert, so dass f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in einem normierten Vektorraum <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>V<\/mi><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> heisst <span class=\"ecbx-1095\">beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt<\/span>, falls es ein <math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt, so dass <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wie in \u00dcbung <a href=\"..\/..\/chapter\/reellwertige-funktionen#x1-92002r38\">3.38<\/a> kann man zeigen, dass die Menge der beschr\u00e4nkten Folgen in <math display=\"inline\"><mi>V<\/mi> <\/math> einen Unterraum des Vektorraums der Folgen in <math display=\"inline\"><mi>V<\/mi> <\/math> bildet.                                                                                                                                                                           <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze32f870511be\"> <a id=\"x1-145007r27\"><\/a> <span class=\"ecbx-1095\">Lemma 5.27 <\/span>(Beschr\u00e4nktheit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Jede konvergente        Folge        in        einem        normierten        Vektorraum<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>V<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> eine konvergente Folge und <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Dann existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daraus folgt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-145008r145\"><\/a> <h4 id=\"z248e123a23ae\" class=\"subsectionHead\"><span class=\"titlemark\">5.3.2 <\/span> <a id=\"x1-1460002\"><\/a>Erste Konsequenzen und Beispiele<\/h4> <p class=\"noindent\">Wir empfehlen den Leserinnen und Lesern sich in diesem Unterabschnitt auf Folgen in <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math> zu konzentrieren. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zac77def40604\"> <a id=\"x1-146001r28\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.28 <\/span>(Konvergente und divergente Folgen in <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Eine konstante Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">konvergiert gegen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Genauso konvergieren schliesslich konstante Folgen gegen den Wert, den sie schliesslich<\/span> <span class=\"ecti-1095\">annehmen.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert gegen Null, das heisst <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Denn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert nach dem Archimedischen Prinzip (Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\"><span class=\"ecti-1095\">2.68<\/span><\/a><span class=\"ecti-1095\">) ein <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>N<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">gilt nun <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>N<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">(und damit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">).<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegeben durch <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ist divergent, da die Folgenglieder <\/span><math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/math> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">hin und her wechseln und sich insbesondere keiner bestimmten Zahl n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hern.<\/span> <p class=\"noindent\"><span class=\"ecti-1095\">Formal argumentiert: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Zahl <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist entweder <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und somit <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">oder <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Im ersten Fall gibt es f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein gerades <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und damit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">(anstatt <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">).<\/span> <span class=\"ecti-1095\">Im zweiten Fall gibt es f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein ungerades <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">(anstatt <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><\/math><span class=\"ecti-1095\">).<\/span><\/p><\/div><\/div> <\/div> <p class=\"indent\">Nach obigem Beispiel k\u00f6nnte man sich die Frage stellen, ob das Konvergenzverhalten einer Folge reeller Zahlen in <math display=\"inline\"><mi>\u2102<\/mi><\/math> dasselbe ist, wenn man die Folge als Folge in <math display=\"inline\"><mi>\u211d<\/mi><\/math> betrachtet. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6796bd396be6\"> <a id=\"x1-146002r29\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.29 <\/span>(Reelle Grenzwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente Folge in <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">reell ist.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Nehmen                          Sie                          an,                          dass<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und                                                w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen                                                Sie<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">so,                 dass                 der                 Ball                 von                 Radius<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">um<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">die reelle Zahlengerade nicht schneidet.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wie schon bei der Stetigkeit von Funktionen m\u00f6chten wir auch hier nicht jedesmal \u201evon Hand\u201c mit&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und&nbsp;<math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math> Grenzwerte berechnen m\u00fcssen. Dazu ist folgende Proposition hilfreich. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7552e1a0c190\"> <a id=\"x1-146003r30\"><\/a> <span class=\"ecbx-1095\">Proposition 5.30 <\/span>(Additive und multiplikative Eigenschaften des Grenzwerts)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">zwei konvergente<\/span> <span class=\"ecti-1095\">Folgen in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist konvergent und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist konvergent und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die Folge <\/span><math display=\"inline\"><mi>\u03b1<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <span class=\"ecti-1095\">die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere bildet die Menge der konvergenten Folgen in<\/span> <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>\u2115<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">einen<\/span> <span class=\"ecti-1095\">Unterraum und der Grenzwert stellt eine lineare Abbildung von diesem Unterraum nach<\/span> <math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">dar.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir setzen <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">F\u00fcr (i) sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> und <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Sei <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach der Dreiecksungleichung ist f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was die Aussage in (i) impliziert. <\/p><p class=\"indent\">F\u00fcr (ii) bemerken wir zuerst, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und m\u00f6chten die letzteren beiden Terme einzeln absch\u00e4tzen. Dabei m\u00fcssen wir sicher stellen, dass <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr grosse <math display=\"inline\"><mi>n<\/mi><\/math> nicht zu gross wird (siehe auch Lemma&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145007r27\">5.27<\/a>). Sei <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> (\u00e4hnlich wie in (i)) so gew\u00e4hlt, dass f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> }<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dann gilt insbesondere <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Damit ist f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was nach obiger Absch\u00e4tzung f\u00fcr <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> die Aussage in (ii) beweist. <\/p><p class=\"indent\">Die Behauptung in (i) und (ii) implizieren auch die letzte Aussage in der Proposition, womit nur noch (iii) zu beweisen ist. Also angenommen <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>A<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir sehen also, dass wir erzwingen k\u00f6nnen, dass <math display=\"inline\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>A<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/math> klein ist, wenn <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> klein ist. Dazu m\u00fcssen wir allerdings verhindern, dass                                                                                                                                                                           <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> zu klein wird. F\u00fcr den formalen Beweis sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Nach Definition von <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> gilt dann nach der umgekehrten Dreiecksungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Also wird <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nicht zu klein und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>A<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2215<\/mo><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2215<\/mo><mn>2<\/mn><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2215<\/mo><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was zu zeigen war. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za0e7c6506469\"> <a id=\"x1-146007r31\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.31.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Vergleichen Sie die Argumente f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Proposition <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94008r50\"><span class=\"ecti-1095\">3.50<\/span><\/a> <span class=\"ecti-1095\">mit dem Beweis von (i) und (ii) in<\/span> <span class=\"ecti-1095\">Proposition <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\"><span class=\"ecti-1095\">5.30<\/span><\/a><span class=\"ecti-1095\">. Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren Sie auch, welche der bewiesenen Aussagen auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r normierte<\/span> <span class=\"ecti-1095\">Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume gelten und wieso.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6b3dd61cd42d\"> <a id=\"x1-146008r32\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.32 <\/span>(Rationale Funktionen als Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie folgende Grenzwerte, wenn sie existieren:<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mn>7<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mn>5<\/mn><\/mrow> <mrow><mn>3<\/mn><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>0<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Formulieren und beweisen Sie allgemeine Versionen von den Beispielen in (i).<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Verwenden Sie hier und auch sonst kein fr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">her erlerntes Kochrezept, das Sie nicht begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nden<\/span> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Stattdessen erweitern Sie mit <\/span><math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein geeignetes <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und argumentieren Sie unter Verwendung des obigen Wissens (Beispiel <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146001r28\"><span class=\"ecti-1095\">5.28<\/span><\/a> <span class=\"ecti-1095\">und Proposition<\/span> <a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\"><span class=\"ecti-1095\">5.30<\/span><\/a><span class=\"ecti-1095\">).<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Eine konvergente Folge in <math display=\"inline\"><mi>\u2102<\/mi><\/math> (oder allgemeiner in einem normierten Vektorraum) mit Grenzwert Null wird auch eine <span class=\"ecbx-1095\">Nullfolge<\/span> genannt. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7cd4c5c0d5ec\"> <a id=\"x1-146011r33\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.33 <\/span>(Nullfolgen und Divergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine komplex-wertige Folge mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegen <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">konvergiert. Zeigen Sie, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">divergiert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5e70c81ba8c7\"> <a id=\"x1-146012r34\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.34 <\/span>(Geometrische Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bezeichnen wir als <\/span><span class=\"ecbi-1095\">geometrische Folge <\/span><span class=\"ecti-1095\">zum Skalierungsfaktor<\/span> <math display=\"inline\"><mi>q<\/mi><\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">untersuchen nun diese geometrische Folge auf Konvergenz.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">wissen wir bereits, dass die Folge <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">divergiert (also keinen Grenzwert hat).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Allgemeiner gilt, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und divergiert ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und daher divergent.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen noch (iii)-(v) beweisen. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii) argumentieren wir indirekt. Angenommen<\/span> <math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>A<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">nach Proposition <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-146003r30\"><span class=\"ecti-1095\">5.30<\/span><\/a> <span class=\"ecti-1095\">und Lemma <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145005r25\"><span class=\"ecti-1095\">5.25<\/span><\/a><span class=\"ecti-1095\">. Dies impliziert<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und wegen<\/span> <math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">, dass<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Da aber<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gilt,<\/span> <span class=\"ecti-1095\">kann <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">nicht der Grenzwert der Folge sein.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iv) sei nun <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">. Wir zeigen, dass die Folge<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist, womit (iv)<\/span> <span class=\"ecti-1095\">aus Lemma <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145007r27\"><span class=\"ecti-1095\">5.27<\/span><\/a> <span class=\"ecti-1095\">folgt. Sei <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nach dem Archimedischen Prinzip (Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\"><span class=\"ecti-1095\">2.68<\/span><\/a><span class=\"ecti-1095\">) existiert ein<\/span> <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>N<\/mi><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>M<\/mi><\/math><span class=\"ecti-1095\">. Nun ergibt die<\/span> <span class=\"ecti-1095\">Bernoulli-Ungleichung (Lemma <\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-79001r5\"><span class=\"ecti-1095\">3.5<\/span><\/a><span class=\"ecti-1095\">) <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>N<\/mi><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">womit die Behauptung gezeigt ist.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (v) sei <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Falls<\/span> <math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">so ist<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Sei nun<\/span> <math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Da<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">, existiert wegen<\/span> <span class=\"ecti-1095\">(iv) ein <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">, so<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>N<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\ud835\udf00<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Somit<\/span> <span class=\"ecti-1095\">gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z39d274814479\"> <a id=\"x1-146018r35\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.35.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>n<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Zeigen Sie zuerst mit Hilfe der Bernoulli-Ungleichung, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z47fddf9b4b99\"> <a id=\"x1-146019r36\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.36 <\/span>(Ces\u00e0ro-Mittel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">konvergente Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Folge der <\/span><span class=\"ecbi-1095\">Ces<\/span><span class=\"ecbi-1095\">\u00e0<\/span><span class=\"ecbi-1095\">ro-Mittel <\/span><span class=\"ecti-1095\">(auch <\/span><span class=\"ecbi-1095\">arithmetische Mittel <\/span><span class=\"ecti-1095\">oder <\/span><span class=\"ecbi-1095\">Cauchy-Mittel<\/span> <span class=\"ecti-1095\">genannt) <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegeben durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">konvergiert und<\/span> <span class=\"ecti-1095\">denselben Grenzwert wie <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">hat.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berzeugen Sie sich auch davon, dass die umgekehrte Implikation nicht gilt, das heisst, dass die<\/span> <span class=\"ecti-1095\">Konvergenz der Ces<\/span><span class=\"ecti-1095\">\u00e0<\/span><span class=\"ecti-1095\">ro-Mittel nicht Konvergenz der Folge impliziert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z94b0969d646d\"> <a id=\"x1-146020r37\"><\/a> <span class=\"ecbx-1095\">Applet 5.37 <\/span>(Einige Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"391px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/FgDxWNhj\/width\/688\/height\/391\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten verschiedene Folgen und k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen mittels Verkleinern der <\/span><math display=\"inline\"><mi>x<\/mi><\/math><span class=\"ecti-1095\">-Achse<\/span> <span class=\"ecti-1095\">die Konvergenz- und Divergenzeigenschaften der Folgen beobachten.<\/span> <\/p> <\/div> <a id=\"x1-146021r146\"><\/a> <h4 id=\"z7dae186e48c0\" class=\"subsectionHead\"><span class=\"titlemark\">5.3.3 <\/span> <a id=\"x1-1470003\"><\/a>Teilfolgen<\/h4> <p class=\"noindent\">Oft m\u00f6chte man anstelle einer Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nur einen \u201e Teil\u201c der Folge betrachten, wobei wir im Gegensatz zur Indexverschiebung in Lemma&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145005r25\">5.25<\/a> manchmal auch unendlich viele Folgenglieder wegstreichen wollen. Dies ist insbesondere dann der Fall, wenn die Folge nicht konvergiert. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd2983185910b\"> <a id=\"x1-147001r38\"><\/a> <span class=\"ecbx-1095\">Definition 5.38 <\/span>(Teilfolge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Wenn <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> eine                       Folge                       in                       einer                       Menge <math display=\"inline\"><mi>X<\/mi><\/math> ist                                                                                                                  und <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> eine streng         monoton         wachsende         Folge         ist,         dann         wird <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> eine                                                  <span class=\"ecbx-1095\">Teilfolge                                           <\/span>von <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> genannt. <\/p> <\/div> <p class=\"indent\">Liegt eine Teilfolge einer konvergenten Folge vor, so konvergiert diese gegen denselben Grenzwert wie die Folge. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zac3a5a4dfdef\"> <a id=\"x1-147002r39\"><\/a> <span class=\"ecbx-1095\">Lemma 5.39 <\/span>(Konvergenz von Teilfolgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente Folge in einem metrischen Raum <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Jede Teilfolge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">konvergiert und hat denselben Grenzwert <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8dfc21f50b29\"> <a id=\"x1-147003r40\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.40.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Lemma <\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-147002r39\"><span class=\"ecti-1095\">5.39<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r          eine          streng          monoton          wachsende          Folge<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>k<\/mi><\/math> <span class=\"ecti-1095\">(wieso?).<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Eine Folge kann konvergente Teilfolgen besitzen, ohne selbst zu konvergieren. Beispielsweise hat die Folge <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die konvergente (konstante) Teilfolge <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> konvergiert aber nicht, wie wir schon gesehen haben. In der Tat haben wir mit Lemma&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-147002r39\">5.39<\/a> jetzt ein k\u00fcrzeres Argument. Falls die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> gegen <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> konvergieren w\u00fcrde, so m\u00fcssten die beiden konstanten Folgen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> auch gegen <math display=\"inline\"><mi>A<\/mi><\/math> konvergieren. Dies ist nat\u00fcrlich nicht m\u00f6glich, da die eine gegen                                                                                                                                                                           <math display=\"inline\"><mn>1<\/mn><\/math> und die andere gegen <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> konvergiert. <\/p><p class=\"indent\">In gewissen Situationen l\u00e4sst sich aus dem Konvergenzverhalten von Teilfolgen trotzdem etwas \u00fcber das Konvergenzverhalten der gesamten Folge sagen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z74a7f2112239\"> <a id=\"x1-147004r41\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.41 <\/span>(Teilfolgen von Teilfolgen und Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge in <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">genau dann gegen <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">konvergiert, wenn jede Teilfolge von <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Teilfolge besitzt, die gegen <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Betrachten                                        Sie                                        zu<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die                                                                                                      Menge<\/span> <math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo> <mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>\ud835\udf00<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und zeigen Sie indirekt, dass diese endlich sein muss.<\/span><\/p><\/details>  <\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9c09931fda6c\"> <a id=\"x1-147005r42\"><\/a> <span class=\"ecbx-1095\">Proposition 5.42 <\/span>(H\u00e4ufungspunkte einer Folge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge in<\/span> <span class=\"ecti-1095\">einem metrischen Raum <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Ein Punkt <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">heisst<\/span> <span class=\"ecbi-1095\">H<\/span><span class=\"ecbi-1095\">\u00e4<\/span><span class=\"ecbi-1095\">ufungspunkt <\/span><span class=\"ecti-1095\">von <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">falls die folgenden <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalenten Bedingungen erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt sind.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gibt eine Teilfolge <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gibt es ein <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen (a) gilt. Sei also <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> eine konvergente Teilfolge von <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> mit Grenzwert <math display=\"inline\"><mi>A<\/mi><\/math> und sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert ein <math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>K<\/mi><\/math><\/span><span class=\"period\">.<\/span> Sei nun <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>K<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann erf\u00fcllt <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> die Bedingung <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> wie gewollt und (b) ist erf\u00fcllt. <\/p><p class=\"indent\">Angenommen (b) gilt. Wir m\u00f6chten rekursiv eine Teilfolge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> finden mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Diese konvergiert dann gegen <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> da f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> die Ungleichung <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math>                                                                                                                                                                           f\u00fcr alle <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\ud835\udf00<\/mi><\/mrow><\/mfrac><\/math> erf\u00fcllt ist. <\/p><p class=\"indent\">Sei <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gibt es ein <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Nun nehmen wir an, dass <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> bereits konstruiert sind mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u2113<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>k<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir setzen <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert nach Voraussetzung ein <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies beendet den Induktionsschritt und wir erhalten durch Rekursion die gew\u00fcnschte Teilfolge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> mit Grenzwert <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z13973e47d1cc\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Obiger                                                                                                    Beweis ist formal nicht ganz unproblematisch. Denn zum Unterschied von der Rekursion, welche wir am Ende von Abschnitt <a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-510001\">2.2.1<\/a> besprochen haben, m\u00fcssen wir hier eigentlich eine Wahl f\u00fcr <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> treffen. Da diese Wahl nicht nur einmal notwendig ist, sondern abz\u00e4hlbar oft, so haben wir in obiger Formulierung eigentlich eine (schwache) Version des Auswahlaxioms verwendet. Wir werden uns diese Freiheit hier und auch in \u00e4hnlichen Situationen erlauben, ohne dieses Auswahlaxiom  genauer  zu  besprechen.  Mit  ein  Grund  daf\u00fcr  ist,  dass  ein  Grossteil  der modernen Mathematik an diesem Auswahlaxiom der Axiomatischen Mengenlehre gebunden ist. Es ist aber auch m\u00f6glich (wenn auch anstrengend) die Verwendung dieses Auswahlaxioms in obigem Beweis zu vermeiden indem man bei jeder Wahl sicherstellt, dass man das minimale <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit allen gew\u00fcnschten Eigenschaften verwendet. <\/p> <\/div> <a id=\"x1-147008r147\"><\/a> <h4 id=\"z899045bf5002\" class=\"subsectionHead\"><span class=\"titlemark\">5.3.4 <\/span> <a id=\"x1-1480004\"><\/a>Konvergenz in endlich-dimensionalen Vektorr\u00e4umen<\/h4> <p class=\"noindent\">Im Allgemeinen h\u00e4ngt der Konvergenzbegriff auf einer Menge <math display=\"inline\"><mi>X<\/mi><\/math> von der Metrik ab, die man auf <math display=\"inline\"><mi>X<\/mi><\/math> betrachtet. Folgende \u00dcbung enth\u00e4lt ein Beispiel. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf7f1abe21ca1\"> <a id=\"x1-148001r43\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.43 <\/span>(Manhattan und SNCF sind sehr verschieden)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Finden Sie eine Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die zwar bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Manhattanmetrik, aber nicht bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der franz<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sischen Eisenbahnmetrik<\/span> <span class=\"ecti-1095\">konvergiert (wobei wir <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und damit <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit einer Teilmenge von <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">identifizieren).<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr normierte endlich-dimensionale Vektorr\u00e4ume ist die Situation oft vorteilshafter. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z31334a5bac0a\"> <a id=\"x1-148002r44\"><\/a> <span class=\"ecbx-1095\">Proposition 5.44.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">, sei<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi> <\/mrow>  <\/math> <span class=\"ecti-1095\">eine Folge<\/span> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">, und<\/span> <span class=\"ecti-1095\">sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Folgende Aussagen sind <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> <span class=\"ecti-1095\">konvergiert gegen <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> <span class=\"ecti-1095\">konvergiert gegen <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> <span class=\"ecti-1095\">konvergiert gegen <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> <span class=\"ecti-1095\">konvergiert die Folge der Komponenten <\/span><math display=\"inline\"><msub><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><\/mrow><\/msub><mo fence=\"true\" form=\"postfix\">)<\/mo><mo fence=\"true\" form=\"postfix\">)<\/mo> <mrow><mi>n<\/mi><\/mrow><\/math> <span class=\"ecti-1095\">gegen <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Inbesondere gilt diese <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenz auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Folge in<\/span> <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">In der Tat werden wir sp\u00e4ter sehen, dass man in obiger Proposition eine beliebige Norm auf <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> betrachen kann. Auf Grund von Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a> werden wir oft von Konvergenz einer Folge in <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> oder                                                                                                                                                                           <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> sprechen, ohne die Norm anzugeben. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir beweisen zuerst die \u00c4quivalenz der Aussagen in (i), (ii), und (iii) und verwenden daf\u00fcr die Ungleichungen <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>d<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;und<\/mtext><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-148007r2\" \/><mstyle class=\"maketag\"><mtext>(5.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msqrt><mrow><mi>d<\/mi><\/mrow><\/msqrt><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-148008r3\" \/><mstyle class=\"maketag\"><mtext>(5.3)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> In der Tat behauptet die erste Ungleichung (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148007r2\">5.2<\/a>) bloss, dass der maximale Absolutbetrag kleiner gleich der Summe der Absolutbetr\u00e4ge, und die Summe der Absolutbetr\u00e4ge kleiner gleich <math display=\"inline\"><mi>d<\/mi><\/math> mal dem maximalen Absolutbetrages ist. Die Ungleichung (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148008r3\">5.3<\/a>) ergibt sich analog aus <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msubsup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msubsup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>d<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msubsup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mi>\u221e<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span>                                                                                                                                                                           <\/p><p class=\"indent\">Unter Verwendung der Ungleichungen in (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148007r2\">5.2<\/a>) und (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148008r3\">5.3<\/a>) ist der Beweis der \u00c4quivalenz der Aussagen in (i), (ii) und (iii) ziemlich direkt. Zur Illustration beweisen wir (i) <math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><\/math> (ii); alle anderen Implikationen verfiziert man analog. Sei also <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und sei <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi> <\/mrow>  <mo class=\"MathClass-bin\">\u2212<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mi>d<\/mi><\/mrow><\/mfrac><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> (wir verwenden hier, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi> <\/mrow>  <\/math> nach Annahme bez\u00fcglich der Norm <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> gegen <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math> konvergiert). Nach (<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148007r2\">5.2<\/a>) gilt f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>d<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was (i) <math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <\/math> (ii) beweist. <\/p><p class=\"indent\">Zum Schluss beweisen wir nun die \u00c4quivalenz der Aussagen in (i) und (iv). Angenommen (i) gilt. Somit gibt es f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi> <\/mrow>  <mo class=\"MathClass-bin\">\u2212<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere gilt f\u00fcr <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">,<\/span> womit <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> gegen <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert, da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war. <\/p><p class=\"indent\">Wir nehmen nun umgekehrt (iv) an, also dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi> <\/mrow>  <\/math> gegen <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert f\u00fcr jedes <span class=\"maperiod\"><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gibt es zu <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> ein <math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi> <\/mrow>  <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>N<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Sei <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dann gilt f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mi>n<\/mi><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig ist, folgt die Konvergenz von <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mrow><mi>n<\/mi><\/mrow><\/math> gegen <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><\/math><\/span><span class=\"period\">,<\/span> was das Lemma beweist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb59352108958\"> <a id=\"x1-148009r45\"><\/a> <span class=\"ecbx-1095\">Korollar 5.45 <\/span>(Reduktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine komplexwertige Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist genau dann konvergent (mit Grenzwert <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><span class=\"ecti-1095\">),<\/span> <span class=\"ecti-1095\">wenn die beiden reellwertigen Folgen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent sind (mit Grenzwerten <\/span><math display=\"inline\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir identifizieren <math display=\"inline\"><mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> (gewissermassen tautologisch) via der Bijektion <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dann ist <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> womit eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>\u2102<\/mi><\/math> genau dann gegen <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> konvergiert, wenn <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> gegen <math display=\"inline\"><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert. Hiermit folgt das Korollar in der Tat aus Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a>(iv). <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2d7d2a07a5c2\"> <a id=\"x1-148010r46\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.46.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert und geben Sie den Grenzwert an. Impliziert umgekehrt die Konvergenz von <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">die Konvergenz von <\/span><span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"endquote\">?<\/span> <\/p> <\/div> <p class=\"indent\">Folgende \u00dcbung gibt ein Beispiel eines unendlich-dimensionalen Vektorraums <math display=\"inline\"><mi>V<\/mi> <\/math> und zweier Normen auf <span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">,<\/span> die einen unterschiedlichen Konvergenzbegriff definieren. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9a6667fbc069\"> <a id=\"x1-148011r47\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.47 <\/span>(<math display=\"inline\"><mn>1<\/mn><\/math>-Norm und Konvergenz im Mittel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ein kompaktes<\/span> <span class=\"ecti-1095\">Intervall mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">. Wir betrachten<\/span> <span class=\"ecti-1095\">den Vektorraum <\/span><math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>C<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit den Normen <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">definiert in Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-1380002\"><span class=\"ecti-1095\">5.1.2<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass Konvergenz <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <span class=\"ecti-1095\">auch die Konvergenz <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">impliziert.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie eine Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>V<\/mi> <\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn> <\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Konvergenz bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">nennt man auch gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssige Konvergenz; wir werden diese sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter in diesem Semester<\/span> <span class=\"ecti-1095\">nochmals einf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren und genauer untersuchen. Konvergenz bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der Norm<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">nennt<\/span> <span class=\"ecti-1095\">man auch Konvergenz im Mittel. Diese <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung hat also gezeigt, dass Konvergenz im Mittel und<\/span> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssige Konvergenz verschiedene Begriffe sind.<\/span> <\/p> <\/div> <a id=\"x1-148014r144\"><\/a> \n","protected":false},"author":1089,"menu_order":3,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-64","chapter","type-chapter","status-publish","hentry"],"part":61,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/64","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/64\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/64\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=64"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=64"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=64"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=64"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}