{"id":74,"date":"2021-12-15T09:53:15","date_gmt":"2021-12-15T09:53:15","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-6\/"},"modified":"2021-12-15T09:53:15","modified_gmt":"2021-12-15T09:53:15","slug":"weitere-lernmaterialien-6","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-6\/","title":{"raw":"Weitere Lernmaterialien","rendered":"Weitere Lernmaterialien"},"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=\"z77a0fef12d9e\" class=\"sectionHead\"><span class=\"titlemark\">6.7 <\/span> <a id=\"x1-1830007\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-183001r181\"><\/a> <h4 id=\"z542ce8e82b8a\" class=\"subsectionHead\"><span class=\"titlemark\">6.7.1 <\/span> <a id=\"x1-1840001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Dieses Kapitel stellt die Grundlagen f\u00fcr Konvergenzbetrachtungen bereit, wobei wir auch einige elementare Grenzwerte bereits berechnen konnten. Die Berechnung dieser Grenzwerte erforderte mit unserem derzeitigen Wissen noch sehr viel Geschick, was mit Hilfe der Regel von de l\u2019H\u00f4pital sp\u00e4ter erheblich einfacher werden wird. Das heisst, dass die wichtigsten Resultate dieses Kapitels nicht durch die konkreten Beispiele oder auch die ersten speziellen Berechnungsmethoden gegeben sind, sondern vielmehr durch die folgenden S\u00e4tze: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Konvergenzverhalten f\u00fcr monotone Folgen in Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-158001r5\">6.5<\/a>. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Definition und Eigenschaften von Limes Superior in Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-159004r11\">6.11<\/a> (und analog f\u00fcr Limes Inferior). <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die Existenz von konvergenten Teilfolgen in Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\">6.15<\/a>. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Der Begriff der Cauchy-Folge und das Cauchy-Kriterium in Satz <a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a>. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Sandwich-Lemma f\u00fcr Folgen und Funktionen.<\/div><\/div> <p class=\"noindent\">Auch wichtig f\u00fcr sp\u00e4tere Berechnungen wird der Zusammenhang zwischen Folgenkonvergenz und Stetigkeit in Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a> des vorherigen Kapitels sein. Wir bemerken noch, dass Limes Superior und Limes Inferior n\u00fctzliche allgemeine Werkzeuge sind, die mitunter auch im Beweis der Konvergenz einer Folge auftreten k\u00f6nnen. Denn <math display=\"inline\"><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\">\u2208<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> und <math display=\"inline\"><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-rel\">\u2208<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> k\u00f6nnen f\u00fcr jede reellwertige 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> betrachtet werden, und das Erf\u00fcllen der Gleichung <math display=\"inline\"><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><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-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ist nach Korollar <a href=\"..\/..\/chapter\/reelle-folgen#x1-159011r14\">6.14<\/a> zur Konvergenz der Folge \u00e4quivalent. Dies ist vergleichbar damit, dass (wie zum Beispiel im Beweis von Korollar <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-101001r71\">3.71<\/a> \u00fcber das Maximum von stetigen Funktionen) das Supremum im Beweis f\u00fcr die Existenz eines Maximums wichtig sein kann. <\/p><p class=\"indent\">Wir haben insgesamt <math display=\"inline\"><mn>6<\/mn><\/math> Konvergenzen f\u00fcr Funktionen ausf\u00fchrlich definiert, wobei es aber insgesamt                                                                                                                                                                           <math display=\"inline\"><mn>1<\/mn><mn>5<\/mn><\/math> Kombinationen (wieso?) f\u00fcr reellwertige Funktionen <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> zu definieren g\u00e4be. Betrachten wir Teilmengen <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> reellwertige Funktionen <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> und Punkte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> so gibt es nochmals drei M\u00f6glichkeiten (reelle oder uneigentliche Grenzwerte). F\u00fcr komplexwertige Funktionen gibt es noch eine weitere Definition f\u00fcr jede der m\u00f6glichen Bewegungen. Es w\u00e4re wohl eher langweilig, all diese Definitionen einzeln auszuformulieren und ihre (jeweils sehr analogen) Eigenschaften aufzulisten. Sie sollten sich aber \u00fcber die verschiedenen M\u00f6glichkeiten und Definitionen bewusst sein, siehe aber folgendes Applet. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5e94883b9222\"> <a id=\"x1-184001r54\"><\/a> <span class=\"ecbx-1095\">Applet 6.54 <\/span>(40 Definitionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 660px\"><iframe height=\"414px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/hqfsh3y9\/width\/660\/height\/414\/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 fassen  alle  (und  einige  weitere)  Definitionen  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  Konvergenz  in  diesem  Applet<\/span> <span class=\"ecti-1095\">zusammen. Versuchen Sie die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">hnlichkeiten und Unterschiede der verschiedenen Definitionen<\/span> <span class=\"ecti-1095\">zu finden.<\/span> <\/p> <\/div> <p class=\"indent\">Des Weiteren konnten wir die reelle Exponentialfunktion mit einem Grenzwert definieren, welche wir gemeinsam mit der Logarithmusfunktion und allgemeinen Potenzen ab nun mit den gewohnten Eigenschaften verwenden d\u00fcrfen. Die Definition der Exponentialfunktion hat auch zu der Ungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/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\">f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> (siehe Abschnitt <a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1670002\">6.3.2<\/a>) gef\u00fchrt, welche f\u00fcr einige Grenzwertberechnungen n\u00fctzlich war. <\/p><p class=\"indent\">Auch haben wir gesehen, dass das Riemann-Integral sich als ein Grenzwert der sogenannten Riemann-Summen auffassen l\u00e4sst, was auch zu einer Definition eines vektorwertigen Riemann-Integrals gef\u00fchrt hat. <a id=\"x1-184002r184\"><\/a> <\/p> <h4 id=\"zae161fc81c5b\" class=\"subsectionHead\"><span class=\"titlemark\">6.7.2 <\/span> <a id=\"x1-1850002\"><\/a>\u00dcbungen<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb3998b169d93\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Stetige Fortsetzung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">stetige Funktionen mit der Eigenschaft, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>\u211a<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>\u211a<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf500daa38afe\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Eine Umkehrung des Zwischenwertsatzes)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/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> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">Intervall zu <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/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-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion, die folgende Eigenschaften erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt:<\/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 alle <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist das Urbild <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">abgeschlossen.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt den Zwischenwertsatz, das heisst, f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gibt es ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb266f95d5f4b\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie die Ungleichung <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/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\">In der Tat werden wir sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter eine explizite Form der Asymptotik von <\/span><math display=\"inline\"> <mfrac><mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">(das Gesetz von Stirling) sehen, welche diese Ungleichung versch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rft.<\/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     Induktion     und     die     Tatsache,     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\">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><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">monoton wachsend ist.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd3fddc601d0f\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(H\u00e4ufungspunkte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><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>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass folgende drei Aussagen <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent sind.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt der Menge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt einer injektiven 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 Folgengliedern <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/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> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist der Grenzwert einer injektiven 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 Folgengliedern <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/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><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z269758ffa8cf\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Abgeschlossene Menge der H\u00e4ufungspunkte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Menge der H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkte einer reellwertigen Folge (oder einer<\/span> <span class=\"ecti-1095\">Teilmenge <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">eine abgeschlossene Teilmenge von <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bildet.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zeef5f66075c0\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Landau Notation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nden Sie, inwiefern die Gleichungen zu<\/span> <math display=\"inline\"><mi>k<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>k<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2113<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>\u2113<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">Sinn ergeben. Verwenden Sie dies, um die Asymptotik f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">von<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mfrac><mrow><mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>5<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow> <mrow><mn>5<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>7<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mn>7<\/mn><\/mrow> <mrow><mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>8<\/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\"><span class=\"ecti-1095\">sowie<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>5<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow> <mrow><mn>5<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>7<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mn>7<\/mn><\/mrow> <mrow><mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>8<\/mn><\/mrow><\/mfrac> <\/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\">zu beschreiben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze99ad862ceb0\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Gross- und Klein-Omega)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien zwei Funktionen <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">auf einer Teilmenge <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gegeben und sei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Definieren Sie in Analogie zur Definition von Gross-O und Klein-o die Beziehungen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03a9<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mstyle class=\"text\"><mtext>&nbsp;f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/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\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c9<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mstyle class=\"text\"><mtext>&nbsp;f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/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> <p class=\"noindent\"><span class=\"ecti-1095\">welche zum Ausdruck bringen, dass <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">in der N<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">he von <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch ein<\/span> <span class=\"ecti-1095\">positives Vielfaches von <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist respektive dass <\/span><math display=\"inline\"><mfrac><mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gegen Null geht f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir wollen in der n\u00e4chsten \u00dcbung den Zusammenhang zwischen \u201eunseren axiomatisch eingef\u00fchrten reellen Zahlen\u201c und den \u201ereellen Zahlen als Steigung von quasi-linearen Abbildungen\u201c von Abschnitt <a href=\"#x1-3010005\">A.2.5<\/a> besprechen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z6812b7a72fbd\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Steigungen von quasi-linearen Abbildungen)<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\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">eine quasi-lineare Abbildung wie in Abschnitt <\/span><a href=\"#x1-3010005\"><span class=\"ecti-1095\">A.2.5<\/span><\/a><span class=\"ecti-1095\">. Zeigen Sie, dass<\/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><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>n<\/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\"><span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">existiert.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><\/math> <span class=\"ecti-1095\">die additive Gruppe der quasi-linearen Abbildungen. Zeigen Sie, dass die Abbildung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03a8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><mo class=\"MathClass-rel\">\u21a6<\/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><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/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\">ein Homomorphismus ist und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03a8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udca6<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Konstruieren Sie zu jedem <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine quasi-lineare Abbildung <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><\/math> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><mi>\u03a8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><\/math><span class=\"ecti-1095\">quasi-linear<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><mi>\u03a8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udca6<\/mi><\/math> <span class=\"ecti-1095\">nur endlich viele Werte annimmt.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Zusammen sehen wir also in der Tat, dass <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi mathvariant=\"bold-script\">\ud835\udca6<\/mi><\/math> <span class=\"ecti-1095\">als abelsche Gruppe isomorph zu <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist. Mit etwas mehr Arbeit l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich beweisen, dass in der Tat ein K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rperisomorphismus<\/span> <span class=\"ecti-1095\">vorliegt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z36c4b1ef2eaf\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Cauchy-Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie direkt, dass eine Folge im <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">genau dann eine Cauchy-Folge ist, wenn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><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> <span class=\"ecti-1095\">die reelle Folge der <\/span><math display=\"inline\"><mi>j<\/mi><\/math><span class=\"ecti-1095\">-ten<\/span> <span class=\"ecti-1095\">Komponenten eine Cauchy-Folge ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5bf59aba42f7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Bilder von Cauchy-Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine                                              Teilmenge                                              und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig         stetige         Funktion.         Zeigen         Sie,         dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">Cauchy-Folgen      auf      Cauchy-Folgen      abbildet      (f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r      jede      Cauchy-Folge<\/span> <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> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><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\">ist                                                                                                         auch<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/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> <span class=\"ecti-1095\">eine Cauchy-Folge). Gilt dies auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Funktionen, die stetig, aber nicht gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig stetig<\/span> <span class=\"ecti-1095\">sind?<\/span> <\/p> <\/div> <p class=\"indent\">Unter Verwendung von Folgen und Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\">6.15<\/a> lassen sich viele Aussagen aus Kapitel <a href=\"..\/..\/part\/funktionen-und-die-reellen-zahlen#x1-760003\">3<\/a> anders beweisen, was wir in den folgenden \u00dcbung illustrieren m\u00f6chten. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z005fc2aa448f\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Beschr\u00e4nktheit mit Hilfe von Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/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-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion<\/span> <span class=\"ecti-1095\">auf einem kompakten Intervall <\/span><math display=\"inline\"><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\">zu <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><span class=\"ecti-1095\">. Wir wollen<\/span> <span class=\"ecti-1095\">zeigen, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Gehen Sie per Widerspruch vor und finden Sie eine Folge <\/span><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> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><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\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/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-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>n<\/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> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wenden Sie nun Satz <\/span><a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\"><span class=\"ecti-1095\">6.15<\/span><\/a> <span class=\"ecti-1095\">an.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z490b8ace2eb7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Gleichm\u00e4ssige Stetigkeit mit Hilfe von Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/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-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion auf einem kompakten Intervall <\/span><math display=\"inline\"><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\">zu <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Gehen Sie nach  einem  <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlichen  Prinzip  vor  wie  in  obiger  <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung,  um  zu  zeigen,  dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig stetig ist.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> \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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\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=\"z77a0fef12d9e\" class=\"sectionHead\"><span class=\"titlemark\">6.7 <\/span> <a id=\"x1-1830007\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-183001r181\"><\/a> <h4 id=\"z542ce8e82b8a\" class=\"subsectionHead\"><span class=\"titlemark\">6.7.1 <\/span> <a id=\"x1-1840001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Dieses Kapitel stellt die Grundlagen f\u00fcr Konvergenzbetrachtungen bereit, wobei wir auch einige elementare Grenzwerte bereits berechnen konnten. Die Berechnung dieser Grenzwerte erforderte mit unserem derzeitigen Wissen noch sehr viel Geschick, was mit Hilfe der Regel von de l\u2019H\u00f4pital sp\u00e4ter erheblich einfacher werden wird. Das heisst, dass die wichtigsten Resultate dieses Kapitels nicht durch die konkreten Beispiele oder auch die ersten speziellen Berechnungsmethoden gegeben sind, sondern vielmehr durch die folgenden S\u00e4tze: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Konvergenzverhalten f\u00fcr monotone Folgen in Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-158001r5\">6.5<\/a>. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Definition und Eigenschaften von Limes Superior in Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-159004r11\">6.11<\/a> (und analog f\u00fcr Limes Inferior). <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die Existenz von konvergenten Teilfolgen in Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\">6.15<\/a>. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Der Begriff der Cauchy-Folge und das Cauchy-Kriterium in Satz <a href=\"..\/..\/chapter\/cauchy-folgen#x1-163001r26\">6.26<\/a>. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Sandwich-Lemma f\u00fcr Folgen und Funktionen.<\/div><\/div> <p class=\"noindent\">Auch wichtig f\u00fcr sp\u00e4tere Berechnungen wird der Zusammenhang zwischen Folgenkonvergenz und Stetigkeit in Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a> des vorherigen Kapitels sein. Wir bemerken noch, dass Limes Superior und Limes Inferior n\u00fctzliche allgemeine Werkzeuge sind, die mitunter auch im Beweis der Konvergenz einer Folge auftreten k\u00f6nnen. Denn <math display=\"inline\"><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\">\u2208<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> und <math display=\"inline\"><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-rel\">\u2208<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> k\u00f6nnen f\u00fcr jede reellwertige 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> betrachtet werden, und das Erf\u00fcllen der Gleichung <math display=\"inline\"><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><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-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ist nach Korollar <a href=\"..\/..\/chapter\/reelle-folgen#x1-159011r14\">6.14<\/a> zur Konvergenz der Folge \u00e4quivalent. Dies ist vergleichbar damit, dass (wie zum Beispiel im Beweis von Korollar <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-101001r71\">3.71<\/a> \u00fcber das Maximum von stetigen Funktionen) das Supremum im Beweis f\u00fcr die Existenz eines Maximums wichtig sein kann. <\/p><p class=\"indent\">Wir haben insgesamt <math display=\"inline\"><mn>6<\/mn><\/math> Konvergenzen f\u00fcr Funktionen ausf\u00fchrlich definiert, wobei es aber insgesamt                                                                                                                                                                           <math display=\"inline\"><mn>1<\/mn><mn>5<\/mn><\/math> Kombinationen (wieso?) f\u00fcr reellwertige Funktionen <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> zu definieren g\u00e4be. Betrachten wir Teilmengen <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> reellwertige Funktionen <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> und Punkte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> so gibt es nochmals drei M\u00f6glichkeiten (reelle oder uneigentliche Grenzwerte). F\u00fcr komplexwertige Funktionen gibt es noch eine weitere Definition f\u00fcr jede der m\u00f6glichen Bewegungen. Es w\u00e4re wohl eher langweilig, all diese Definitionen einzeln auszuformulieren und ihre (jeweils sehr analogen) Eigenschaften aufzulisten. Sie sollten sich aber \u00fcber die verschiedenen M\u00f6glichkeiten und Definitionen bewusst sein, siehe aber folgendes Applet. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5e94883b9222\"> <a id=\"x1-184001r54\"><\/a> <span class=\"ecbx-1095\">Applet 6.54 <\/span>(40 Definitionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 660px\"><iframe height=\"414px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/hqfsh3y9\/width\/660\/height\/414\/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 fassen  alle  (und  einige  weitere)  Definitionen  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  Konvergenz  in  diesem  Applet<\/span> <span class=\"ecti-1095\">zusammen. Versuchen Sie die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">hnlichkeiten und Unterschiede der verschiedenen Definitionen<\/span> <span class=\"ecti-1095\">zu finden.<\/span> <\/p> <\/div> <p class=\"indent\">Des Weiteren konnten wir die reelle Exponentialfunktion mit einem Grenzwert definieren, welche wir gemeinsam mit der Logarithmusfunktion und allgemeinen Potenzen ab nun mit den gewohnten Eigenschaften verwenden d\u00fcrfen. Die Definition der Exponentialfunktion hat auch zu der Ungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/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\">f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> (siehe Abschnitt <a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1670002\">6.3.2<\/a>) gef\u00fchrt, welche f\u00fcr einige Grenzwertberechnungen n\u00fctzlich war. <\/p><p class=\"indent\">Auch haben wir gesehen, dass das Riemann-Integral sich als ein Grenzwert der sogenannten Riemann-Summen auffassen l\u00e4sst, was auch zu einer Definition eines vektorwertigen Riemann-Integrals gef\u00fchrt hat. <a id=\"x1-184002r184\"><\/a> <\/p> <h4 id=\"zae161fc81c5b\" class=\"subsectionHead\"><span class=\"titlemark\">6.7.2 <\/span> <a id=\"x1-1850002\"><\/a>\u00dcbungen<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb3998b169d93\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Stetige Fortsetzung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">stetige Funktionen mit der Eigenschaft, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>\u211a<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>\u211a<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf500daa38afe\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Eine Umkehrung des Zwischenwertsatzes)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/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> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">Intervall zu <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/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-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion, die folgende Eigenschaften erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt:<\/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 alle <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist das Urbild <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">abgeschlossen.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt den Zwischenwertsatz, das heisst, f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gibt es ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb266f95d5f4b\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie die Ungleichung <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/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\">In der Tat werden wir sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter eine explizite Form der Asymptotik von <\/span><math display=\"inline\"> <mfrac><mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">(das Gesetz von Stirling) sehen, welche diese Ungleichung versch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rft.<\/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     Induktion     und     die     Tatsache,     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\">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><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">monoton wachsend ist.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd3fddc601d0f\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(H\u00e4ufungspunkte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><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>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass folgende drei Aussagen <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent sind.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt der Menge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt einer injektiven 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 Folgengliedern <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/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> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist der Grenzwert einer injektiven 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 Folgengliedern <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/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><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z269758ffa8cf\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Abgeschlossene Menge der H\u00e4ufungspunkte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Menge der H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkte einer reellwertigen Folge (oder einer<\/span> <span class=\"ecti-1095\">Teilmenge <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">eine abgeschlossene Teilmenge von <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bildet.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zeef5f66075c0\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Landau Notation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nden Sie, inwiefern die Gleichungen zu<\/span> <math display=\"inline\"><mi>k<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>k<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2113<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>\u2113<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">Sinn ergeben. Verwenden Sie dies, um die Asymptotik f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">von<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mfrac><mrow><mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>5<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow> <mrow><mn>5<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>7<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mn>7<\/mn><\/mrow> <mrow><mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>8<\/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\"><span class=\"ecti-1095\">sowie<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>5<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow> <mrow><mn>5<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>7<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mn>7<\/mn><\/mrow> <mrow><mn>3<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>8<\/mn><\/mrow><\/mfrac> <\/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\">zu beschreiben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze99ad862ceb0\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Gross- und Klein-Omega)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien zwei Funktionen <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">auf einer Teilmenge <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gegeben und sei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Definieren Sie in Analogie zur Definition von Gross-O und Klein-o die Beziehungen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03a9<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mstyle class=\"text\"><mtext>&nbsp;f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/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\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c9<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mstyle class=\"text\"><mtext>&nbsp;f\u00fcr&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/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> <p class=\"noindent\"><span class=\"ecti-1095\">welche zum Ausdruck bringen, dass <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">in der N<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">he von <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch ein<\/span> <span class=\"ecti-1095\">positives Vielfaches von <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist respektive dass <\/span><math display=\"inline\"><mfrac><mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gegen Null geht f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir wollen in der n\u00e4chsten \u00dcbung den Zusammenhang zwischen \u201eunseren axiomatisch eingef\u00fchrten reellen Zahlen\u201c und den \u201ereellen Zahlen als Steigung von quasi-linearen Abbildungen\u201c von Abschnitt <a href=\"#x1-3010005\">A.2.5<\/a> besprechen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6812b7a72fbd\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Steigungen von quasi-linearen Abbildungen)<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\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">eine quasi-lineare Abbildung wie in Abschnitt <\/span><a href=\"#x1-3010005\"><span class=\"ecti-1095\">A.2.5<\/span><\/a><span class=\"ecti-1095\">. Zeigen Sie, dass<\/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><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>n<\/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\"><span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">existiert.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><\/math> <span class=\"ecti-1095\">die additive Gruppe der quasi-linearen Abbildungen. Zeigen Sie, dass die Abbildung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03a8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><mo class=\"MathClass-rel\">\u21a6<\/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><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/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\">ein Homomorphismus ist und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03a8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udca6<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Konstruieren Sie zu jedem <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine quasi-lineare Abbildung <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><\/math> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><mi>\u03a8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><\/math><span class=\"ecti-1095\">quasi-linear<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><mi>\u03a8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udca6<\/mi><\/math> <span class=\"ecti-1095\">nur endlich viele Werte annimmt.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Zusammen sehen wir also in der Tat, dass <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcac<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi mathvariant=\"bold-script\">\ud835\udca6<\/mi><\/math> <span class=\"ecti-1095\">als abelsche Gruppe isomorph zu <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist. Mit etwas mehr Arbeit l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich beweisen, dass in der Tat ein K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rperisomorphismus<\/span> <span class=\"ecti-1095\">vorliegt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z36c4b1ef2eaf\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Cauchy-Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie direkt, dass eine Folge im <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">genau dann eine Cauchy-Folge ist, wenn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><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> <span class=\"ecti-1095\">die reelle Folge der <\/span><math display=\"inline\"><mi>j<\/mi><\/math><span class=\"ecti-1095\">-ten<\/span> <span class=\"ecti-1095\">Komponenten eine Cauchy-Folge ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5bf59aba42f7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Bilder von Cauchy-Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine                                              Teilmenge                                              und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig         stetige         Funktion.         Zeigen         Sie,         dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">Cauchy-Folgen      auf      Cauchy-Folgen      abbildet      (f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r      jede      Cauchy-Folge<\/span> <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> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><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\">ist                                                                                                         auch<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/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> <span class=\"ecti-1095\">eine Cauchy-Folge). Gilt dies auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Funktionen, die stetig, aber nicht gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig stetig<\/span> <span class=\"ecti-1095\">sind?<\/span> <\/p> <\/div> <p class=\"indent\">Unter Verwendung von Folgen und Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\">6.15<\/a> lassen sich viele Aussagen aus Kapitel <a href=\"..\/..\/part\/funktionen-und-die-reellen-zahlen#x1-760003\">3<\/a> anders beweisen, was wir in den folgenden \u00dcbung illustrieren m\u00f6chten. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z005fc2aa448f\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Beschr\u00e4nktheit mit Hilfe von Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/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-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion<\/span> <span class=\"ecti-1095\">auf einem kompakten Intervall <\/span><math display=\"inline\"><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\">zu <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><span class=\"ecti-1095\">. Wir wollen<\/span> <span class=\"ecti-1095\">zeigen, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Gehen Sie per Widerspruch vor und finden Sie eine Folge <\/span><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> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><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\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/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-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>n<\/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> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wenden Sie nun Satz <\/span><a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\"><span class=\"ecti-1095\">6.15<\/span><\/a> <span class=\"ecti-1095\">an.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z490b8ace2eb7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Gleichm\u00e4ssige Stetigkeit mit Hilfe von Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/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-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion auf einem kompakten Intervall <\/span><math display=\"inline\"><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\">zu <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Gehen Sie nach  einem  <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlichen  Prinzip  vor  wie  in  obiger  <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung,  um  zu  zeigen,  dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig stetig ist.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> 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