{"id":69,"date":"2021-12-15T09:53:13","date_gmt":"2021-12-15T09:53:13","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/cauchy-folgen\/"},"modified":"2021-12-15T09:53:13","modified_gmt":"2021-12-15T09:53:13","slug":"cauchy-folgen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/cauchy-folgen\/","title":{"raw":"Cauchy-Folgen","rendered":"Cauchy-Folgen"},"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=\"zf99c68c53723\" class=\"sectionHead\"><span class=\"titlemark\">6.2 <\/span> <a id=\"x1-1620002\"><\/a>Cauchy-Folgen<\/h3> <p class=\"noindent\">Wir f\u00fchren eine weitere Eigenschaft von Folgen ein. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zb9c0c054b402\"> <a id=\"x1-162001r22\"><\/a> <span class=\"ecbx-1095\">Definition 6.22 <\/span>(Cauchy-Folge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">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> ist eine Cauchy-Folge, 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 <\/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><mi>m<\/mi><\/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-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>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\">Intuitiv ausgedr\u00fcckt ist eine Cauchy-Folge also eine Folge, deren Folgenglieder f\u00fcr grosse Indizes immer n\u00e4her beieinanderliegen. In dieser Formulierung macht die Aussage folgender \u00dcbung Sinn. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z7385a88c34c2\"> <a id=\"x1-162002r23\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.23 <\/span>(Konvergente Folgen sind Cauchy-Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass eine konvergente Folge in einem metrischen Raum eine Cauchy-Folge 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\">Verwenden                Sie                f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                ein                gegebenes<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die                    Definition                    des                    Grenzwerts                    zu<\/span> <math display=\"inline\"><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">und die Dreiecksungleichung.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Ein Ziel dieses Kapitels ist die Umkehrung f\u00fcr reelle Cauchy-Folgen zu zeigen, also dass jede Cauchy-Folge selbst bereits einen Grenzwert besitzt. Dies ist n\u00fctzlich, da wir f\u00fcr den Beweis der Konvergenz nach Definition eigentlich den Grenzwert bereits kennen m\u00fcssen. Wollen wir hingegen zeigen, dass eine Folge eine Cauchy-Folge ist, so m\u00fcssen wir nur die Folgenglieder der Folge betrachten. Damit werden wir im n\u00e4chsten Kapitel viele neue Zahlen und Funktionen definieren k\u00f6nnen.<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Falls wir den Grenzwert f\u00fcr die Konstruktion dieser Funktionen bereits kennen m\u00fcssten, so k\u00f6nnten wir ja damit keine <span class=\"ecti-1095\">neuen <\/span>Funktionen definieren.<\/span><\/span> <\/p><p class=\"indent\">Um Konvergenz einer Cauchy-Folge zu zeigen, kann man folgendes n\u00fctzliches Kriterium verwenden. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z331489b47fc3\"> <a id=\"x1-162003r24\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.24 <\/span>(Konvergente Teilfolgen von Cauchy-Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass eine Cauchy-Folge genau dann konvergiert, wenn sie eine konvergente<\/span> <span class=\"ecti-1095\">Teilfolge besitzt.<\/span> <\/p> <\/div> <p class=\"indent\">Im Sinne dieses Kapitels werden wir uns hier auf Cauchy-Folgen in den reellen Zahlen konzentrieren. Im zweiten Semester werden wir wieder auf Cauchy-Folgen in allgemeinen metrischen R\u00e4umen zu sprechen kommen. Wir bemerken noch, dass sich Cauchy-Folgen in <math display=\"inline\"><mi>\u2102<\/mi><\/math> mittels dem Inhalt dieses Kapitels und folgender \u00dcbung verstehen lassen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z708591337b13\"> <a id=\"x1-162004r25\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.25.<\/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=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann 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 eine Cauchy-Folge, wenn <\/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\">Cauchy-Folgen sind.<\/span> <\/p> <\/div> <a id=\"x1-162005r161\"><\/a> <h4 id=\"zdd38bd30e33d\" class=\"subsectionHead\"><span class=\"titlemark\">6.2.1 <\/span> <a id=\"x1-1630001\"><\/a>Reelle Cauchy-Folgen<\/h4> <p class=\"noindent\">Wie angek\u00fcndigt, konzentrieren wir uns auf reelle Cauchy-Folgen und beweisen folgenden Satz. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"ze8936158fcf2\"> <a id=\"x1-163001r26\"><\/a> <span class=\"ecbx-1095\">Satz 6.26 <\/span>(Cauchy-Kriterium f\u00fcr Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine reelle Folge ist genau dann konvergent, wenn sie eine Cauchy-Folge ist.<\/span> <\/p> <\/div> <p class=\"indent\">Wie erw\u00e4hnt, hat der Begriff der Cauchy-Folge gemeinsam mit Satz <a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> haben gegen\u00fcber der Definition der Konvergenz den entscheidenden Vorteil, dass wir den Grenzwert nicht kennen m\u00fcssen, um zu zeigen, dass eine Folge eine Cauchy-Folge ist (und damit nach Satz&nbsp;<a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> einen Grenzwert besitzt). Des Weiteren hat Satz&nbsp;<a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> gegen\u00fcber Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-158001r5\">6.5<\/a> den Vorteil, dass er nicht nur f\u00fcr spezielle Folgen anwendbar ist. <\/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\">Angenommen <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> ist eine reelle Folge mit <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/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 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 f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> gilt <span class=\"maperiod\"><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><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> gilt somit auch                                                                                                                                                                           <\/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>m<\/mi><\/mrow><\/msub> <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\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/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><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\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><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\">Dies beweist, 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> eine Cauchy-Folge ist. <\/p><p class=\"indent\">Sei nun umgekehrt <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 Cauchy-Folge. F\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> existiert dann 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\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <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\">&lt;<\/mo> <mn>1<\/mn><\/math> f\u00fcr <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> Insbesondere gilt also <\/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\">\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> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/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-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/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 <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher ist <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 beschr\u00e4nkte Folge (wieso? \u2013 siehe auch den Beweis von Lemma <a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145007r27\">5.27<\/a>). Des Weiteren existiert nach Annahme f\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> 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\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <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\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/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\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir setzen <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>N<\/mi><\/math> und erhalten                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/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\">Wir betrachten nun Limes Inferior und Limes Superior der Folge (welche ja nach Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\">6.15<\/a> Grenzwerte konvergenter Teilfolgen sind) und erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> liminf<\/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\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> limsup<\/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\">\u2264<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/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\">Insbesondere gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi class=\"qopname\">liminf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><msub><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da aber <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, erhalten wir Gleichheit von Limes Superior und Limes Inferior und daher Konvergenz der Folge nach Korollar <a href=\"..\/..\/chapter\/reelle-folgen#x1-159011r14\">6.14<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zec2317c3afb8\"> <a id=\"x1-163002r27\"><\/a> <span class=\"ecbx-1095\">Beispiel 6.27 <\/span>(Falsches Kriterium)<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 reelle Folge. Wir behaupten, dass die Bedingung<\/span> <\/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> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/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\"><span class=\"ecti-1095\">nicht zur Konvergenz der Folge <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent ist.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Wir setzen <\/span><math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/math> <span class=\"ecti-1095\">zu einer Folge fort, in dem wir alle rationalen Zahlen zwischen<\/span> <math display=\"inline\"><mi>\u2113<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">mit Nenner<\/span> <math display=\"inline\"><mi>\u2113<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">aufsteigend<\/span> <span class=\"ecti-1095\">auflisten. Falls <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u2113<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/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\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">, dann<\/span> <span class=\"ecti-1095\">haben <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens<\/span> <span class=\"ecti-1095\">Abstand <\/span><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u2113<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Dies<\/span> <span class=\"ecti-1095\">zeigt, dass <\/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><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <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> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">aber <\/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 trotzdem nicht beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und insbesondere nicht konvergent.<\/span> <\/p> <\/div> <p class=\"indent\">Die Aussage von Satz <a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> ist fundamental f\u00fcr die Analysis und ist das, was man in einem allgemeineren Kontext unter Vollst\u00e4ndigkeit der reellen Zahlen versteht (siehe auch Kapitel&nbsp;<span class=\"ecbx-1095\">??<\/span> vom zweiten Semester). In anderen Worten ist Satz <a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> zum Vollst\u00e4ndigkeitsaxiom \u00e4quivalent im Sinne der folgenden \u00dcbung. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9672f55d4448\"> <a id=\"x1-163003r28\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.28 <\/span>(Vollst\u00e4ndigkeit der reellen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten hier erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren, wie aus dem Archimedischen Prinzip und der Aussage von<\/span> <span class=\"ecti-1095\">Satz <\/span><a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\"><span class=\"ecti-1095\">6.26<\/span><\/a> <span class=\"ecti-1095\">das         Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom         folgt.         Genauer         sei<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein                            geordneter                            K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper,                            der<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">als                   dichte                   Teilmenge                   enth<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt                   (im<\/span> <span class=\"ecti-1095\">Sinne von Korollar <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68006r70\"><span class=\"ecti-1095\">2.70<\/span><\/a><span class=\"ecti-1095\">) und in dem alle Cauchy-Folgen konvergent sind. Zeigen Sie, dass<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">das Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/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\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">nicht-leere Teilmengen von <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. Beginnen Sie mit<\/span> <span class=\"ecti-1095\">zwei Punkten <\/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> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math><span class=\"ecti-1095\">, betrachten<\/span> <span class=\"ecti-1095\">Sie den Punkt <\/span><math display=\"inline\"><mfrac><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">und unterscheiden Sie die F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gibt ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac> <mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gibt ein <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Weder (a) noch (b) treffen zu.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Fahren Sie je nach Fall verschieden fort.<\/span><\/p><\/details>  <\/div> <a id=\"x1-163007r163\"><\/a> <h4 id=\"z904631ac6a88\" class=\"subsectionHead\"><span class=\"titlemark\">6.2.2 <\/span> <a id=\"x1-1640002\"><\/a>Ein Diagramm f\u00fcr die Zusammenh\u00e4nge der Begriffe und S\u00e4tze<\/h4> <p class=\"noindent\">Wir fassen obiges Wissen \u00fcber reelle Folgen in folgendem Diagramm zusammen und empfehlen Ihnen sich zu \u00fcberlegen, was genau die einzelnen Pfeile bedeuten und welche S\u00e4tze sie andeuten. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=524&amp;url=Pictures\/folgen\/diagramm.pdf\"><img id=\"za84a03f8c4d7\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/folgen\/diagramm.svg\" width=\"524\"><\/div> <a id=\"x1-164001r2\"><\/a> <a id=\"x1-164002\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;6.2: <\/span><span class=\"content\">Einige wichtige Eigenschaften von reellen Folgen und wichtige               S\u00e4tze, die diese Eigenschaften in Verbindung bringen.                      &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <a id=\"x1-164003r162\"><\/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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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; 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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=\"zf99c68c53723\" class=\"sectionHead\"><span class=\"titlemark\">6.2 <\/span> <a id=\"x1-1620002\"><\/a>Cauchy-Folgen<\/h3> <p class=\"noindent\">Wir f\u00fchren eine weitere Eigenschaft von Folgen ein. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb9c0c054b402\"> <a id=\"x1-162001r22\"><\/a> <span class=\"ecbx-1095\">Definition 6.22 <\/span>(Cauchy-Folge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">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> ist eine Cauchy-Folge, 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 <\/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><mi>m<\/mi><\/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-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>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\">Intuitiv ausgedr\u00fcckt ist eine Cauchy-Folge also eine Folge, deren Folgenglieder f\u00fcr grosse Indizes immer n\u00e4her beieinanderliegen. In dieser Formulierung macht die Aussage folgender \u00dcbung Sinn. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7385a88c34c2\"> <a id=\"x1-162002r23\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.23 <\/span>(Konvergente Folgen sind Cauchy-Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass eine konvergente Folge in einem metrischen Raum eine Cauchy-Folge 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\">Verwenden                Sie                f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                ein                gegebenes<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die                    Definition                    des                    Grenzwerts                    zu<\/span> <math display=\"inline\"><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">und die Dreiecksungleichung.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Ein Ziel dieses Kapitels ist die Umkehrung f\u00fcr reelle Cauchy-Folgen zu zeigen, also dass jede Cauchy-Folge selbst bereits einen Grenzwert besitzt. Dies ist n\u00fctzlich, da wir f\u00fcr den Beweis der Konvergenz nach Definition eigentlich den Grenzwert bereits kennen m\u00fcssen. Wollen wir hingegen zeigen, dass eine Folge eine Cauchy-Folge ist, so m\u00fcssen wir nur die Folgenglieder der Folge betrachten. Damit werden wir im n\u00e4chsten Kapitel viele neue Zahlen und Funktionen definieren k\u00f6nnen.<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Falls wir den Grenzwert f\u00fcr die Konstruktion dieser Funktionen bereits kennen m\u00fcssten, so k\u00f6nnten wir ja damit keine <span class=\"ecti-1095\">neuen <\/span>Funktionen definieren.<\/span><\/span> <\/p><p class=\"indent\">Um Konvergenz einer Cauchy-Folge zu zeigen, kann man folgendes n\u00fctzliches Kriterium verwenden. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z331489b47fc3\"> <a id=\"x1-162003r24\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.24 <\/span>(Konvergente Teilfolgen von Cauchy-Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass eine Cauchy-Folge genau dann konvergiert, wenn sie eine konvergente<\/span> <span class=\"ecti-1095\">Teilfolge besitzt.<\/span> <\/p> <\/div> <p class=\"indent\">Im Sinne dieses Kapitels werden wir uns hier auf Cauchy-Folgen in den reellen Zahlen konzentrieren. Im zweiten Semester werden wir wieder auf Cauchy-Folgen in allgemeinen metrischen R\u00e4umen zu sprechen kommen. Wir bemerken noch, dass sich Cauchy-Folgen in <math display=\"inline\"><mi>\u2102<\/mi><\/math> mittels dem Inhalt dieses Kapitels und folgender \u00dcbung verstehen lassen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z708591337b13\"> <a id=\"x1-162004r25\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.25.<\/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=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann 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 eine Cauchy-Folge, wenn <\/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\">Cauchy-Folgen sind.<\/span> <\/p> <\/div> <a id=\"x1-162005r161\"><\/a> <h4 id=\"zdd38bd30e33d\" class=\"subsectionHead\"><span class=\"titlemark\">6.2.1 <\/span> <a id=\"x1-1630001\"><\/a>Reelle Cauchy-Folgen<\/h4> <p class=\"noindent\">Wie angek\u00fcndigt, konzentrieren wir uns auf reelle Cauchy-Folgen und beweisen folgenden Satz. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze8936158fcf2\"> <a id=\"x1-163001r26\"><\/a> <span class=\"ecbx-1095\">Satz 6.26 <\/span>(Cauchy-Kriterium f\u00fcr Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine reelle Folge ist genau dann konvergent, wenn sie eine Cauchy-Folge ist.<\/span> <\/p> <\/div> <p class=\"indent\">Wie erw\u00e4hnt, hat der Begriff der Cauchy-Folge gemeinsam mit Satz <a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> haben gegen\u00fcber der Definition der Konvergenz den entscheidenden Vorteil, dass wir den Grenzwert nicht kennen m\u00fcssen, um zu zeigen, dass eine Folge eine Cauchy-Folge ist (und damit nach Satz&nbsp;<a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> einen Grenzwert besitzt). Des Weiteren hat Satz&nbsp;<a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> gegen\u00fcber Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-158001r5\">6.5<\/a> den Vorteil, dass er nicht nur f\u00fcr spezielle Folgen anwendbar ist. <\/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\">Angenommen <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> ist eine reelle Folge mit <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/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 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 f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> gilt <span class=\"maperiod\"><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><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> gilt somit auch                                                                                                                                                                           <\/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>m<\/mi><\/mrow><\/msub> <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\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/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><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\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><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\">Dies beweist, 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> eine Cauchy-Folge ist. <\/p><p class=\"indent\">Sei nun umgekehrt <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 Cauchy-Folge. F\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> existiert dann 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\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <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\">&lt;<\/mo> <mn>1<\/mn><\/math> f\u00fcr <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> Insbesondere gilt also <\/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\">\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> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/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-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/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 <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher ist <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 beschr\u00e4nkte Folge (wieso? \u2013 siehe auch den Beweis von Lemma <a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145007r27\">5.27<\/a>). Des Weiteren existiert nach Annahme f\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> 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\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <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\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/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\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir setzen <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>N<\/mi><\/math> und erhalten                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/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\">Wir betrachten nun Limes Inferior und Limes Superior der Folge (welche ja nach Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\">6.15<\/a> Grenzwerte konvergenter Teilfolgen sind) und erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> liminf<\/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\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> limsup<\/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\">\u2264<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/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\">Insbesondere gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi class=\"qopname\">liminf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><msub><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da aber <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, erhalten wir Gleichheit von Limes Superior und Limes Inferior und daher Konvergenz der Folge nach Korollar <a href=\"..\/..\/chapter\/reelle-folgen#x1-159011r14\">6.14<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zec2317c3afb8\"> <a id=\"x1-163002r27\"><\/a> <span class=\"ecbx-1095\">Beispiel 6.27 <\/span>(Falsches Kriterium)<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 reelle Folge. Wir behaupten, dass die Bedingung<\/span> <\/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> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/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\"><span class=\"ecti-1095\">nicht zur Konvergenz der Folge <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent ist.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Wir setzen <\/span><math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/math> <span class=\"ecti-1095\">zu einer Folge fort, in dem wir alle rationalen Zahlen zwischen<\/span> <math display=\"inline\"><mi>\u2113<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">mit Nenner<\/span> <math display=\"inline\"><mi>\u2113<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">aufsteigend<\/span> <span class=\"ecti-1095\">auflisten. Falls <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u2113<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/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\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">, dann<\/span> <span class=\"ecti-1095\">haben <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens<\/span> <span class=\"ecti-1095\">Abstand <\/span><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>\u2113<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Dies<\/span> <span class=\"ecti-1095\">zeigt, dass <\/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><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <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> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">aber <\/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 trotzdem nicht beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und insbesondere nicht konvergent.<\/span> <\/p> <\/div> <p class=\"indent\">Die Aussage von Satz <a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> ist fundamental f\u00fcr die Analysis und ist das, was man in einem allgemeineren Kontext unter Vollst\u00e4ndigkeit der reellen Zahlen versteht (siehe auch Kapitel&nbsp;<span class=\"ecbx-1095\">??<\/span> vom zweiten Semester). In anderen Worten ist Satz <a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a> zum Vollst\u00e4ndigkeitsaxiom \u00e4quivalent im Sinne der folgenden \u00dcbung. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9672f55d4448\"> <a id=\"x1-163003r28\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.28 <\/span>(Vollst\u00e4ndigkeit der reellen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten hier erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren, wie aus dem Archimedischen Prinzip und der Aussage von<\/span> <span class=\"ecti-1095\">Satz <\/span><a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\"><span class=\"ecti-1095\">6.26<\/span><\/a> <span class=\"ecti-1095\">das         Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom         folgt.         Genauer         sei<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein                            geordneter                            K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper,                            der<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">als                   dichte                   Teilmenge                   enth<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt                   (im<\/span> <span class=\"ecti-1095\">Sinne von Korollar <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68006r70\"><span class=\"ecti-1095\">2.70<\/span><\/a><span class=\"ecti-1095\">) und in dem alle Cauchy-Folgen konvergent sind. Zeigen Sie, dass<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">das Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/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\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">nicht-leere Teilmengen von <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. Beginnen Sie mit<\/span> <span class=\"ecti-1095\">zwei Punkten <\/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> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math><span class=\"ecti-1095\">, betrachten<\/span> <span class=\"ecti-1095\">Sie den Punkt <\/span><math display=\"inline\"><mfrac><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">und unterscheiden Sie die F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gibt ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac> <mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es gibt ein <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Weder (a) noch (b) treffen zu.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Fahren Sie je nach Fall verschieden fort.<\/span><\/p><\/details>  <\/div> <a id=\"x1-163007r163\"><\/a> <h4 id=\"z904631ac6a88\" class=\"subsectionHead\"><span class=\"titlemark\">6.2.2 <\/span> <a id=\"x1-1640002\"><\/a>Ein Diagramm f\u00fcr die Zusammenh\u00e4nge der Begriffe und S\u00e4tze<\/h4> <p class=\"noindent\">Wir fassen obiges Wissen \u00fcber reelle Folgen in folgendem Diagramm zusammen und empfehlen Ihnen sich zu \u00fcberlegen, was genau die einzelnen Pfeile bedeuten und welche S\u00e4tze sie andeuten. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=524&amp;url=Pictures\/folgen\/diagramm.pdf\"><img decoding=\"async\" id=\"za84a03f8c4d7\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/folgen\/diagramm.svg\" width=\"524\" \/><\/div> <a id=\"x1-164001r2\"><\/a> <a id=\"x1-164002\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;6.2: <\/span><span class=\"content\">Einige wichtige Eigenschaften von reellen Folgen und wichtige               S\u00e4tze, die diese Eigenschaften in Verbindung bringen.                      &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <a id=\"x1-164003r162\"><\/a> \n","protected":false},"author":1089,"menu_order":2,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-69","chapter","type-chapter","status-publish","hentry"],"part":67,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/69","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\/69\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/67"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/69\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=69"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=69"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=69"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=69"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}