{"id":38,"date":"2021-12-15T09:53:01","date_gmt":"2021-12-15T09:53:01","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/maximum-und-supremum\/"},"modified":"2021-12-15T09:53:01","modified_gmt":"2021-12-15T09:53:01","slug":"maximum-und-supremum","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/maximum-und-supremum\/","title":{"raw":"Maximum und Supremum","rendered":"Maximum und Supremum"},"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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doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z9f72cd9893f8\" class=\"sectionHead\"><span class=\"titlemark\">2.5 <\/span> <a id=\"x1-620005\"><\/a>Maximum und Supremum<\/h3> <a id=\"x1-62001r61\"><\/a> <h4 id=\"zb38cceba105c\" class=\"subsectionHead\"><span class=\"titlemark\">2.5.1 <\/span> <a id=\"x1-630001\"><\/a>Maximum und Minimum<\/h4> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"ze9b5315645a4\"> <a id=\"x1-63001r56\"><\/a> <span class=\"ecbx-1095\">Definition 2.56 <\/span>(Maximum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Wir                                                  sagen,                                                  dass <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> das                            <span class=\"ecbx-1095\">Maximum                        <\/span>einer                            Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ist,                                                                                                                 falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und                                                      f\u00fcr                                                      alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> die                                                                                                      Ungleichung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> gilt. <\/p> <\/div> <p class=\"indent\">Wir d\u00fcrfen in der Tat von <span class=\"underline\">dem<\/span> Maximum einer Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> sprechen, da es durch die Definition eindeutig bestimmt ist. Denn falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> beide die Eigenschaften eines Maximums erf\u00fcllen, so folgt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> (weil <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> ein Maximum ist) und <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> (weil <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein Maximum ist) und damit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Ein abgeschlossenes Intervall <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> mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> hat <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> als                                                                                                                                                                           Maximum. Auch nicht-leere endliche Teilmengen und viele weitere Mengen besitzen ein Maximum. Es gibt jedoch auch Mengen, die kein Maximum besitzen. Beispielsweise hat das offene Intervall <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> mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> kein Maximum (beweisen Sie dies als \u00dcbung) - es w\u00fcrde sich zwar der Endpunkt <math display=\"inline\"><mi>b<\/mi><\/math> anbieten, doch dieser liegt nicht in der Menge <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> und ist also kein Kandidat f\u00fcr das Maximum. Wir werden in K\u00fcrze einen Begriff einf\u00fchren, der auf&nbsp;<math display=\"inline\"><mi>b<\/mi><\/math> zutrifft und gewissermassen als Ersatz f\u00fcr das Maximum angesehen werden kann. <\/p><p class=\"indent\">Des Weiteren kann <math display=\"inline\"><mi>\u211d<\/mi><\/math> (oder auch Intervalle der Form <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math>) kein Maximum besitzen, da f\u00fcr beliebige <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die Ungleichung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> gilt und damit <math display=\"inline\"><mi>x<\/mi><\/math> kein Maximum sein kann. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z46afa81895a6\"> <a id=\"x1-63002r57\"><\/a> <span class=\"ecbx-1095\">Definition 2.57 <\/span>(Minimum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Wir                                                  sagen,                                                  dass <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das                            <span class=\"ecbx-1095\">Minimum                        <\/span>einer                            Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ist,                                                                                                                 falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr                                                                                                                alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> gilt. <\/p> <\/div> <p class=\"indent\">Die obige Diskussion l\u00e4sst sich auf analoge Weise f\u00fcr das Minimum anwenden. Dieses ist also eindeutig bestimmt, muss aber nicht unbedingt existieren. <a id=\"x1-63003r63\"><\/a> <\/p> <h4 id=\"z5c0dc917a9f2\" class=\"subsectionHead\"><span class=\"titlemark\">2.5.2 <\/span> <a id=\"x1-640002\"><\/a>Supremum und Infimum<\/h4> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zf59dafb4b813\"> <a id=\"x1-64001r58\"><\/a> <span class=\"ecbx-1095\">Definition 2.58 <\/span>(Beschr\u00e4nktheit und Schranken)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> heisst <span class=\"ecbx-1095\">von oben beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt<\/span>, falls es ein <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt mit <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ein solches <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nennt man in diesem Fall eine <span class=\"ecbx-1095\">obere Schranke <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Begriffe \u201e <span class=\"ecbx-1095\">von unten beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt<\/span>\u201c und \u201e<span class=\"ecbx-1095\">untere Schranke<\/span>\u201c sind analog definiert. Eine Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> heisst <span class=\"ecbx-1095\">beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt<\/span>, falls sie von oben und von unten beschr\u00e4nkt ist. <\/p> <\/div> <p class=\"indent\">Wie wir bereits bemerkt haben, hat zum Beispiel das Intervall <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> kein Maximum. Es hat aber obere Schranken, <math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><mn>0<\/mn><\/math> ist ein Beispiel. Nat\u00fcrlich ist <math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><mn>0<\/mn><\/math> keine \u201e gute\u201c obere Schranke; <math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><\/math> oder auch <math display=\"inline\"><mn>2<\/mn><\/math> oder <math display=\"inline\"><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> sind kleinere also auch \u201ebessere\u201c obere Schranken. Die absolut beste obere Schranke ist aber durch <math display=\"inline\"><mn>1<\/mn><\/math> gegeben. Denn nach Definition von <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> ist <math display=\"inline\"><mn>1<\/mn><\/math> sicherlich eine obere Schranke und f\u00fcr jede obere Schranke <math display=\"inline\"><mi>s<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <button class=\"hover-trigger\">(Wieso?)<\/button><span class=\"hover-text\"><span class=\"marginpar\">Wir argumentieren indirekt und nehmen an, <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> sei eine obere Schranke. Dann w\u00e4re <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> da <span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> und damit <span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Da aber <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> auch <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <mi>s<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> impliziert, widerspricht dies der Annahme, dass <math display=\"inline\"><mi>s<\/mi><\/math> eine obere Schranke sei.<\/span><\/span> <\/p><p class=\"indent\">Diese Gedanken f\u00fchren gemeinsam mit dem Vollst\u00e4ndigkeitsaxiom (Axiom (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>) in                                                                                                                                                                           Abschnitt&nbsp;<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-470003\">2.1.3<\/a>) zu folgendem grundlegenden Begriff. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z01db550f70a0\"> <a id=\"x1-64002r59\"><\/a> <span class=\"ecbx-1095\">Satz 2.59 <\/span>(Supremum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine von oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte, nicht-leere Teilmenge. Dann gibt es eine <\/span><span class=\"ecbi-1095\">kleinste obere Schranke <\/span><span class=\"ecti-1095\">von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math><span class=\"ecti-1095\">, die auch das<\/span> <span class=\"ecbi-1095\">Supremum <\/span><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">genannt wird. Formal<\/span> <span class=\"ecti-1095\">gelten also f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">folgende Eigenschaften:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(1)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(<\/span><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist eine obere Schranke) <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(2)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(<\/span><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist kleiner gleich jeder oberen Schranke) <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenterweise kann <\/span><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">auch durch (1) und die folgende Bedingung definiert werden:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(2\u2019)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Kleinere Zahlen sind keine oberen Schranken) <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Um diesen wichtigen Begriff noch etwas genauer zu beleuchten, wollen wir vor dem Beweis noch ein paar Bemerkungen machen. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Falls das Maximum <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> existiert, dann ist <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> eine obere Schranke von <math display=\"inline\"><mi>X<\/mi><\/math> und ist vielmehr auch die kleinste obere Schranke, also <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Denn aus <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> folgt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math> f\u00fcr jede obere Schranke <math display=\"inline\"><mi>s<\/mi><\/math> von <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span>                                                                                                                                                                           <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Wenn das Supremum <math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> in <math display=\"inline\"><mi>X<\/mi><\/math> liegt, dann ist <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> da das Supremum eine obere Schranke ist. Also ist das Supremum eine Verallgemeinerung des Maximums einer Menge. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die Formulierung  \u201ekleinste  obere  Schranke\u201c  ist  nat\u00fcrlich  ein  Synonym  f\u00fcr  das Minimum der oberen Schranken und ist dadurch eindeutig bestimmt, falls es existiert.<\/div><\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis von Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">  Nach Annahme ist <math display=\"inline\"><mi>X<\/mi><\/math> nicht-leer und die Menge der oberen Schranken <math display=\"inline\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> ist ebenfalls nicht-leer. Des Weiteren gilt f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math> die Ungleichung <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">.<\/span> Nach dem Vollst\u00e4ndigkeitsaxiom (Axiom (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>) in Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-470003\">2.1.3<\/a>) folgt daher, dass es ein <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt, f\u00fcr das <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">.<\/span> Aus der ersten Ungleichung folgt, dass <math display=\"inline\"><mi>c<\/mi><\/math> eine obere Schranke von <math display=\"inline\"><mi>X<\/mi><\/math> ist. Aus der zweiten Ungleichung folgt, dass <math display=\"inline\"><mi>c<\/mi><\/math> die kleinste obere Schranke von <math display=\"inline\"><mi>X<\/mi><\/math> ist, und daher erf\u00fcllt <math display=\"inline\"><mi>c<\/mi><\/math> sowohl (1) als auch (2). <\/p><p class=\"indent\">Wir zeigen nun, dass das Supremum auch durch (1) und (2\u2019) charakterisiert wird. Also angenommen <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> dann ist <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Daher kann <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math> keine obere Schranke sein und es existiert ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher erf\u00fcllt&nbsp;<math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> auch (2\u2019). <\/p><p class=\"indent\">Erf\u00fcllt <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nun (1) und (2\u2019), so ist <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> eine obere Schranke und daher ist <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> nach Definition von <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span>                                                                                                                                                                           Falls <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> w\u00e4re, dann w\u00e4re <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr ein <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Nach der zweiten Eigenschaft von <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> g\u00e4be es ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> was der Definition von <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> als (kleinste) obere Schranke widerspricht. Deswegen muss <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> gelten und <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ist eindeutig durch die Bedingungen (1) und (2\u2019) bestimmt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z84db73dc1156\"> <a id=\"x1-64005r60\"><\/a> <span class=\"ecbx-1095\">Applet 2.60 <\/span>(Supremum einer beschr\u00e4nkten nicht-leeren Menge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"261px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/fkddrfvn\/width\/688\/height\/261\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten       eine       beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte       nicht-leere       Teilmenge       von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und zwei <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalente Charakterisierungen des Supremums dieser Menge.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis zur Bedienung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">In diesem und manchen der folgenden Applets k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen sie den<\/span> <span class=\"ecti-1095\">dargestellten Ausschnitt vergr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ssern: je nach Ger<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t mit Mausrad, Auf- und Abbewegung mit<\/span> <span class=\"ecti-1095\">zwei Finger auf dem Trackpad, oder auf mobilen Ger<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten mittels Streckbewegungen mit zwei<\/span> <span class=\"ecti-1095\">Finger.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Genauso wie auch andere Konsequenzen des Vollst\u00e4ndigkeitsaxioms, die wir behandeln werden, ist die Existenz des Supremums in der Tat \u00e4quivalent zum Vollst\u00e4ndigkeitsaxiom. In anderen Worten h\u00e4tten wir anstelle von Axiom (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>) einfach die Aussage von Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> fordern k\u00f6nnen. Mehr dazu finden Sie im Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-750002\">2.7.2<\/a>. <\/p><p class=\"indent\">F\u00fcr eine von unten beschr\u00e4nkte, nicht-leere Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> wird die gr\u00f6sste, untere Schranke auch das <span class=\"ecbx-1095\">Infimum <\/span><math display=\"inline\"><mi class=\"qopname\">inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von <math display=\"inline\"><mi>X<\/mi><\/math> genannt. F\u00fcr das Infimum gilt eine \u00e4hnliche Aussage wie in Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a>: <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7e792d2caa74\"> <a id=\"x1-64006r61\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.61 <\/span>(Existenz des Infimums)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Formulieren und beweisen Sie die analoge Aussage zu Satz <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Infimum. Sie<\/span> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen dazu wie  im  Beweis  von  Satz  <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">vorgehen  oder  das  Supremum  der  Teilmenge<\/span> <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine         von         unten         beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte,         nicht-leere         Teilmenge<\/span> <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">betrachten.<\/span> <\/p> <\/div> <p class=\"indent\">Die in obiger \u00dcbung erschienene Notation l\u00e4sst sich verallgemeinern. Sei <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> eine reelle Zahl und seien <math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> zwei Teilmengen. Wir definieren <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi><mi>A<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi><mi>a<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>A<\/mi><mi>B<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi><mi>b<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Es gelten also beispielsweise die Identit\u00e4ten <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>x<\/mi><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mi>A<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Auch gilt <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> mit <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>d<\/mi><\/math><\/span><span class=\"period\">.<\/span> (Wieso?) <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z7189ab3ef25d\"> <a id=\"x1-64007r62\"><\/a> <span class=\"ecbx-1095\">Proposition 2.62 <\/span>(Supremum unter Streckung)<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\">eine nicht-leere, von oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Teilmenge und sei<\/span> <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>c<\/mi><mi>A<\/mi><\/math> <span class=\"ecti-1095\">von<\/span> <span class=\"ecti-1095\">oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Wir empfehlen Ihnen hier, sich die Aussage dieser (genauso wie der n\u00e4chsten) Proposition zuerst am Begriff des Maximums zu veranschaulichen. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math> und somit auch <math display=\"inline\"><mi>c<\/mi><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi><mi>s<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da aber jedes Element von <math display=\"inline\"><mi>c<\/mi><mi>A<\/mi><\/math> von der Form <math display=\"inline\"><mi>c<\/mi><mi>a<\/mi><\/math> f\u00fcr ein <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> ist, erhalten wir, dass <math display=\"inline\"><mi>c<\/mi><mi>s<\/mi><\/math> eine obere Schranke von <math display=\"inline\"><mi>c<\/mi><mi>A<\/mi><\/math> ist und dass <math display=\"inline\"><mi>c<\/mi><mi>A<\/mi><\/math> von oben beschr\u00e4nkt ist. <\/p><p class=\"indent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert nach Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> ein <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>s<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mi>c<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> f\u00fcr welches die Ungleichung <math display=\"inline\"><mi>c<\/mi><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>c<\/mi><mi>s<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math> gilt. Dies zeigt die zweite charakterisierende Eigenschaft des Supremums und wir erhalten                                                                                                                                                                           <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z80ce696d015c\"> <a id=\"x1-64008r63\"><\/a> <span class=\"ecbx-1095\">Proposition 2.63 <\/span>(Supremum unter Summen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwei nicht-leere, von oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Teilmengen von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">von<\/span> <span class=\"ecti-1095\">oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir definieren <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi> <\/mrow> <\/msub> <\/math> und <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi> <\/mrow> <\/msub> <\/math> f\u00fcr alle <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> was <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> und <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math> impliziert. Da aber jedes Element von <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/math> von dieser Form ist, erhalten wir, dass <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msub><\/math> eine obere Schranke von <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/math>                                                                                                                                                                           ist und dass <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/math> von oben beschr\u00e4nkt ist. <\/p><p class=\"indent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert nach Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> ein <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> mit <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> und ein <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac> <mrow> <mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> was wiederum <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math> impliziert. Dies zeigt die zweite charakterisierende Eigenschaft von <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> in Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> und wir erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <span>&nbsp;&nbsp;<\/span><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-64009r64\"><\/a> <h4 id=\"z293a3502e9ec\" class=\"subsectionHead\"><span class=\"titlemark\">2.5.3 <\/span> <a id=\"x1-650003\"><\/a>Uneigentliche Werte, Suprema und Infima<\/h4> <p class=\"noindent\">In diesem Abschnitt wollen wir die Begriffe \u201eSupremum\u201c und \u201eInfimum\u201c auf beliebige Teilmengen von <math display=\"inline\"><mi>\u211d<\/mi><\/math> erweitern (ohne die in Abschnitt&nbsp;<a href=\"..\/..\/chapter\/maximum-und-supremum#x1-640002\">2.5.2<\/a> getroffenen Annahmen). Dazu verwenden wir die Symbole <math display=\"inline\"><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> die keine reellen Zahlen darstellen. Wir definieren die <span class=\"ecbx-1095\">erweiterte Zahlengerade <\/span>(die auch <span class=\"ecbx-1095\">Zweipunktkompaktifizierung<\/span> von <math display=\"inline\"><mi>\u211d<\/mi><\/math> genannt wird) durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2294<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/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\">und stellen uns diese als die Zahlengerade <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=625&amp;url=Pictures\/Reelle_Zahlen\/Supremum\/uneig_sup.pdf\"><img id=\"z50ef4f81a6cd\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Reelle_Zahlen\/Supremum\/uneig_sup.svg\" width=\"625\"><\/div>  <\/div> <p class=\"noindent\">vor. Hier haben wir den Punkt <math display=\"inline\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/math> rechts von <math display=\"inline\"><mi>\u211d<\/mi><\/math> und den Punkt <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> links von <math display=\"inline\"><mi>\u211d<\/mi><\/math> zu der Gerade hinzugef\u00fcgt. Formaler formuliert: wir erweitern die Relation (Ordnung) <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> auf <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <\/math> gilt, aber keine weiteren <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math>-Relationen f\u00fcr die Symbole <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> erf\u00fcllt sind. Inbesondere schreiben wir auch <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z14e0617f0b27\"> <a id=\"x1-65001r64\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.64 <\/span>(Geometrie der Zweipunktkompaktifizierung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Abbildung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi>x<\/mi><\/mrow><\/mfrac> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/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\">bijektiv ist und die Ordnung erh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt. Das heisst, f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Erweitern Sie<\/span> <math display=\"inline\"><mi>\u03d5<\/mi><\/math> <span class=\"ecti-1095\">zu einer ordnungserhaltenden<\/span> <span class=\"ecti-1095\">Bijektion <\/span><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03d5<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-punc\">:<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren Sie damit das obige Bild der erweiterten Zahlengerade.<\/span> <\/p> <\/div> <p class=\"indent\">Das Maximum und das Minimum einer Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <\/math> ist nun wie in Abschnitt <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-630001\">2.5.1<\/a> definiert (falls es existiert). <\/p><p class=\"indent\">Falls <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> nicht von oben beschr\u00e4nkt ist, dann definieren wir <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>X<\/mi><\/math> leer ist, setzen wir <math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2205<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> (da jedes <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> eine obere Schranke von <math display=\"inline\"><mi>\u2205<\/mi><\/math> darstellt). Analog definieren wir <math display=\"inline\"><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u2205<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> nicht von unten beschr\u00e4nkt ist. <\/p><p class=\"indent\">Folgende \u00dcbungen stellen nat\u00fcrliche Eigenschaften von Supremum und Infimum dar. Sie sollten mindestens eine dieser \u00dcbungen ausarbeiten. Betrachten Sie hierbei die Spezialf\u00e4lle, die zu einem uneigentlichen Supremum oder Infimum f\u00fchren, getrennt und gehen Sie anschliessend wie im Beweis von Proposition <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64008r63\">2.63<\/a> vor. <\/p><p class=\"indent\">Weiter definieren wir f\u00fcr die \u00dcbungen die \u201eRechenregeln\u201c                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mi>\u221e<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>\u221e<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi> <\/mtd> <mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mtd><\/mtr> <\/mtable> <\/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\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>\u221e<\/mi><mo class=\"MathClass-bin\">\u22c5<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <\/mtd> <\/mtr><mtr><mtd class=\"array\" columnalign=\"center\"> <mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <\/mtd> <\/mtr><mtr><mtd class=\"array\" columnalign=\"center\"> <mi>\u221e<\/mi><mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi> <\/mtd> <\/mtr><mtr><mtd class=\"array\" columnalign=\"center\"><mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/mtd><\/mtr> <\/mtable> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> wovon wir einen Teil verwenden werden. Die Ausdr\u00fccke <math display=\"inline\"><mi>\u221e<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><\/math> und <math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>\u221e<\/mi><\/math> oder \u00e4hnliche bleiben wohlgemerkt aber undefiniert. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb12bd7b9fd1c\"> <a id=\"x1-65002r65\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.65 <\/span>(Eigenschaften von Supremum und Infimum unter Vereinigung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">Teilmengen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>Y<\/mi> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Formulieren und beweisen Sie eine analoge Formel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Infimum von<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z0fffa68e9585\"> <a id=\"x1-65003r66\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.66 <\/span>(Eigenschaften von Supremum und Infimum unter Summen und Produkten)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwei nicht-leere Teilmengen. Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><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\">und dass, falls <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Suchen Sie des Weiteren <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnliche Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Infimum.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z19c74f816cec\"> <a id=\"x1-65004r67\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.67.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">eine nicht-leere<\/span> <span class=\"ecti-1095\">Teilmenge von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><table id=\"zd79f609be996\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">Hierbei ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">das Bild von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">unter dem Absolutbetrag <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">(als Funktion von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <a id=\"x1-65005r65\"><\/a> <h4 id=\"z12efc5883900\" class=\"subsectionHead\"><span class=\"titlemark\">2.5.4 <\/span> <a id=\"x1-660004\"><\/a>Verwendung des Supremums und des Infimums<\/h4> <p class=\"noindent\">Das Supremum ist eine nat\u00fcrliche und notwendige Verallgemeinerung des Maximums einer Menge, da letzteres sogar f\u00fcr beschr\u00e4nkte Intervalle nicht existieren muss. Das Supremum kann aber auch hilfreich sein in Situationen, wo das Maximum existiert. Denn falls man beweisen will, dass ein Maximum existiert, dann hat man mit dem Supremum den richtigen Kandidaten und kann den Beweis mit der Existenz des Supremums beginnen. Auf die gleiche Weise ist das Infimum einer Menge eine Verallgemeinerung des Minimums. <\/p><p class=\"indent\">Es ist wichtig, dass Sie sich die charakterisierenden Eigenschaften des Supremums und Infimums einpr\u00e4gen, da diese Begriffe fundamentale Bausteine unserer zu entwickelnden Theorie sein werden. Zum Beispiel werden wir das Integral einer Funktion durch ein Supremum definieren (siehe Figur&nbsp;<a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4008r2\">1.2<\/a> und Kapitel&nbsp;<a href=\"..\/..\/part\/das-riemann-integral#x1-1060004\">4<\/a>).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-66001r62\"><\/a> <\/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.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z9f72cd9893f8\" class=\"sectionHead\"><span class=\"titlemark\">2.5 <\/span> <a id=\"x1-620005\"><\/a>Maximum und Supremum<\/h3> <a id=\"x1-62001r61\"><\/a> <h4 id=\"zb38cceba105c\" class=\"subsectionHead\"><span class=\"titlemark\">2.5.1 <\/span> <a id=\"x1-630001\"><\/a>Maximum und Minimum<\/h4> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze9b5315645a4\"> <a id=\"x1-63001r56\"><\/a> <span class=\"ecbx-1095\">Definition 2.56 <\/span>(Maximum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Wir                                                  sagen,                                                  dass <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> das                            <span class=\"ecbx-1095\">Maximum                        <\/span>einer                            Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ist,                                                                                                                 falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und                                                      f\u00fcr                                                      alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> die                                                                                                      Ungleichung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> gilt. <\/p> <\/div> <p class=\"indent\">Wir d\u00fcrfen in der Tat von <span class=\"underline\">dem<\/span> Maximum einer Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> sprechen, da es durch die Definition eindeutig bestimmt ist. Denn falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> beide die Eigenschaften eines Maximums erf\u00fcllen, so folgt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> (weil <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> ein Maximum ist) und <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> (weil <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein Maximum ist) und damit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Ein abgeschlossenes Intervall <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> mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> hat <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> als                                                                                                                                                                           Maximum. Auch nicht-leere endliche Teilmengen und viele weitere Mengen besitzen ein Maximum. Es gibt jedoch auch Mengen, die kein Maximum besitzen. Beispielsweise hat das offene Intervall <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> mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> kein Maximum (beweisen Sie dies als \u00dcbung) &#8211; es w\u00fcrde sich zwar der Endpunkt <math display=\"inline\"><mi>b<\/mi><\/math> anbieten, doch dieser liegt nicht in der Menge <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> und ist also kein Kandidat f\u00fcr das Maximum. Wir werden in K\u00fcrze einen Begriff einf\u00fchren, der auf&nbsp;<math display=\"inline\"><mi>b<\/mi><\/math> zutrifft und gewissermassen als Ersatz f\u00fcr das Maximum angesehen werden kann. <\/p><p class=\"indent\">Des Weiteren kann <math display=\"inline\"><mi>\u211d<\/mi><\/math> (oder auch Intervalle der Form <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math>) kein Maximum besitzen, da f\u00fcr beliebige <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die Ungleichung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> gilt und damit <math display=\"inline\"><mi>x<\/mi><\/math> kein Maximum sein kann. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z46afa81895a6\"> <a id=\"x1-63002r57\"><\/a> <span class=\"ecbx-1095\">Definition 2.57 <\/span>(Minimum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Wir                                                  sagen,                                                  dass <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das                            <span class=\"ecbx-1095\">Minimum                        <\/span>einer                            Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ist,                                                                                                                 falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr                                                                                                                alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> gilt. <\/p> <\/div> <p class=\"indent\">Die obige Diskussion l\u00e4sst sich auf analoge Weise f\u00fcr das Minimum anwenden. Dieses ist also eindeutig bestimmt, muss aber nicht unbedingt existieren. <a id=\"x1-63003r63\"><\/a> <\/p> <h4 id=\"z5c0dc917a9f2\" class=\"subsectionHead\"><span class=\"titlemark\">2.5.2 <\/span> <a id=\"x1-640002\"><\/a>Supremum und Infimum<\/h4> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf59dafb4b813\"> <a id=\"x1-64001r58\"><\/a> <span class=\"ecbx-1095\">Definition 2.58 <\/span>(Beschr\u00e4nktheit und Schranken)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> heisst <span class=\"ecbx-1095\">von oben beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt<\/span>, falls es ein <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt mit <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ein solches <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nennt man in diesem Fall eine <span class=\"ecbx-1095\">obere Schranke <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Begriffe \u201e <span class=\"ecbx-1095\">von unten beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt<\/span>\u201c und \u201e<span class=\"ecbx-1095\">untere Schranke<\/span>\u201c sind analog definiert. Eine Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> heisst <span class=\"ecbx-1095\">beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt<\/span>, falls sie von oben und von unten beschr\u00e4nkt ist. <\/p> <\/div> <p class=\"indent\">Wie wir bereits bemerkt haben, hat zum Beispiel das Intervall <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> kein Maximum. Es hat aber obere Schranken, <math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><mn>0<\/mn><\/math> ist ein Beispiel. Nat\u00fcrlich ist <math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><mn>0<\/mn><\/math> keine \u201e gute\u201c obere Schranke; <math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><\/math> oder auch <math display=\"inline\"><mn>2<\/mn><\/math> oder <math display=\"inline\"><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> sind kleinere also auch \u201ebessere\u201c obere Schranken. Die absolut beste obere Schranke ist aber durch <math display=\"inline\"><mn>1<\/mn><\/math> gegeben. Denn nach Definition von <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> ist <math display=\"inline\"><mn>1<\/mn><\/math> sicherlich eine obere Schranke und f\u00fcr jede obere Schranke <math display=\"inline\"><mi>s<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <button class=\"hover-trigger\">(Wieso?)<\/button><span class=\"hover-text\"><span class=\"marginpar\">Wir argumentieren indirekt und nehmen an, <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> sei eine obere Schranke. Dann w\u00e4re <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> da <span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> und damit <span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Da aber <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> auch <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <mi>s<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/math> impliziert, widerspricht dies der Annahme, dass <math display=\"inline\"><mi>s<\/mi><\/math> eine obere Schranke sei.<\/span><\/span> <\/p><p class=\"indent\">Diese Gedanken f\u00fchren gemeinsam mit dem Vollst\u00e4ndigkeitsaxiom (Axiom (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>) in                                                                                                                                                                           Abschnitt&nbsp;<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-470003\">2.1.3<\/a>) zu folgendem grundlegenden Begriff. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z01db550f70a0\"> <a id=\"x1-64002r59\"><\/a> <span class=\"ecbx-1095\">Satz 2.59 <\/span>(Supremum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine von oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte, nicht-leere Teilmenge. Dann gibt es eine <\/span><span class=\"ecbi-1095\">kleinste obere Schranke <\/span><span class=\"ecti-1095\">von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math><span class=\"ecti-1095\">, die auch das<\/span> <span class=\"ecbi-1095\">Supremum <\/span><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">genannt wird. Formal<\/span> <span class=\"ecti-1095\">gelten also f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">folgende Eigenschaften:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(1)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(<\/span><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist eine obere Schranke) <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(2)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(<\/span><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist kleiner gleich jeder oberen Schranke) <\/span><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenterweise kann <\/span><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">auch durch (1) und die folgende Bedingung definiert werden:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(2\u2019)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Kleinere Zahlen sind keine oberen Schranken) <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Um diesen wichtigen Begriff noch etwas genauer zu beleuchten, wollen wir vor dem Beweis noch ein paar Bemerkungen machen. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Falls das Maximum <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> existiert, dann ist <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> eine obere Schranke von <math display=\"inline\"><mi>X<\/mi><\/math> und ist vielmehr auch die kleinste obere Schranke, also <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Denn aus <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> folgt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math> f\u00fcr jede obere Schranke <math display=\"inline\"><mi>s<\/mi><\/math> von <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span>                                                                                                                                                                           <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Wenn das Supremum <math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> in <math display=\"inline\"><mi>X<\/mi><\/math> liegt, dann ist <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> da das Supremum eine obere Schranke ist. Also ist das Supremum eine Verallgemeinerung des Maximums einer Menge. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die Formulierung  \u201ekleinste  obere  Schranke\u201c  ist  nat\u00fcrlich  ein  Synonym  f\u00fcr  das Minimum der oberen Schranken und ist dadurch eindeutig bestimmt, falls es existiert.<\/div><\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis von Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">  Nach Annahme ist <math display=\"inline\"><mi>X<\/mi><\/math> nicht-leer und die Menge der oberen Schranken <math display=\"inline\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> ist ebenfalls nicht-leer. Des Weiteren gilt f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math> die Ungleichung <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">.<\/span> Nach dem Vollst\u00e4ndigkeitsaxiom (Axiom (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>) in Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-470003\">2.1.3<\/a>) folgt daher, dass es ein <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt, f\u00fcr das <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">.<\/span> Aus der ersten Ungleichung folgt, dass <math display=\"inline\"><mi>c<\/mi><\/math> eine obere Schranke von <math display=\"inline\"><mi>X<\/mi><\/math> ist. Aus der zweiten Ungleichung folgt, dass <math display=\"inline\"><mi>c<\/mi><\/math> die kleinste obere Schranke von <math display=\"inline\"><mi>X<\/mi><\/math> ist, und daher erf\u00fcllt <math display=\"inline\"><mi>c<\/mi><\/math> sowohl (1) als auch (2). <\/p><p class=\"indent\">Wir zeigen nun, dass das Supremum auch durch (1) und (2\u2019) charakterisiert wird. Also angenommen <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> dann ist <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Daher kann <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math> keine obere Schranke sein und es existiert ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher erf\u00fcllt&nbsp;<math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> auch (2\u2019). <\/p><p class=\"indent\">Erf\u00fcllt <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nun (1) und (2\u2019), so ist <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> eine obere Schranke und daher ist <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> nach Definition von <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span>                                                                                                                                                                           Falls <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> w\u00e4re, dann w\u00e4re <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr ein <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Nach der zweiten Eigenschaft von <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> g\u00e4be es ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> was der Definition von <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> als (kleinste) obere Schranke widerspricht. Deswegen muss <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> gelten und <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ist eindeutig durch die Bedingungen (1) und (2\u2019) bestimmt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z84db73dc1156\"> <a id=\"x1-64005r60\"><\/a> <span class=\"ecbx-1095\">Applet 2.60 <\/span>(Supremum einer beschr\u00e4nkten nicht-leeren Menge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"261px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/fkddrfvn\/width\/688\/height\/261\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten       eine       beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte       nicht-leere       Teilmenge       von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und zwei <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalente Charakterisierungen des Supremums dieser Menge.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis zur Bedienung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">In diesem und manchen der folgenden Applets k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen sie den<\/span> <span class=\"ecti-1095\">dargestellten Ausschnitt vergr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ssern: je nach Ger<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t mit Mausrad, Auf- und Abbewegung mit<\/span> <span class=\"ecti-1095\">zwei Finger auf dem Trackpad, oder auf mobilen Ger<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten mittels Streckbewegungen mit zwei<\/span> <span class=\"ecti-1095\">Finger.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Genauso wie auch andere Konsequenzen des Vollst\u00e4ndigkeitsaxioms, die wir behandeln werden, ist die Existenz des Supremums in der Tat \u00e4quivalent zum Vollst\u00e4ndigkeitsaxiom. In anderen Worten h\u00e4tten wir anstelle von Axiom (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>) einfach die Aussage von Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> fordern k\u00f6nnen. Mehr dazu finden Sie im Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-750002\">2.7.2<\/a>. <\/p><p class=\"indent\">F\u00fcr eine von unten beschr\u00e4nkte, nicht-leere Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> wird die gr\u00f6sste, untere Schranke auch das <span class=\"ecbx-1095\">Infimum <\/span><math display=\"inline\"><mi class=\"qopname\">inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von <math display=\"inline\"><mi>X<\/mi><\/math> genannt. F\u00fcr das Infimum gilt eine \u00e4hnliche Aussage wie in Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a>: <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7e792d2caa74\"> <a id=\"x1-64006r61\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.61 <\/span>(Existenz des Infimums)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Formulieren und beweisen Sie die analoge Aussage zu Satz <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Infimum. Sie<\/span> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen dazu wie  im  Beweis  von  Satz  <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">vorgehen  oder  das  Supremum  der  Teilmenge<\/span> <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine         von         unten         beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte,         nicht-leere         Teilmenge<\/span> <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">betrachten.<\/span> <\/p> <\/div> <p class=\"indent\">Die in obiger \u00dcbung erschienene Notation l\u00e4sst sich verallgemeinern. Sei <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> eine reelle Zahl und seien <math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> zwei Teilmengen. Wir definieren <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi><mi>A<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi><mi>a<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>A<\/mi><mi>B<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi><mi>b<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Es gelten also beispielsweise die Identit\u00e4ten <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>x<\/mi><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mi>A<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Auch gilt <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> mit <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>d<\/mi><\/math><\/span><span class=\"period\">.<\/span> (Wieso?) <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7189ab3ef25d\"> <a id=\"x1-64007r62\"><\/a> <span class=\"ecbx-1095\">Proposition 2.62 <\/span>(Supremum unter Streckung)<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\">eine nicht-leere, von oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Teilmenge und sei<\/span> <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>c<\/mi><mi>A<\/mi><\/math> <span class=\"ecti-1095\">von<\/span> <span class=\"ecti-1095\">oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Wir empfehlen Ihnen hier, sich die Aussage dieser (genauso wie der n\u00e4chsten) Proposition zuerst am Begriff des Maximums zu veranschaulichen. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><\/math> und somit auch <math display=\"inline\"><mi>c<\/mi><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi><mi>s<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da aber jedes Element von <math display=\"inline\"><mi>c<\/mi><mi>A<\/mi><\/math> von der Form <math display=\"inline\"><mi>c<\/mi><mi>a<\/mi><\/math> f\u00fcr ein <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> ist, erhalten wir, dass <math display=\"inline\"><mi>c<\/mi><mi>s<\/mi><\/math> eine obere Schranke von <math display=\"inline\"><mi>c<\/mi><mi>A<\/mi><\/math> ist und dass <math display=\"inline\"><mi>c<\/mi><mi>A<\/mi><\/math> von oben beschr\u00e4nkt ist. <\/p><p class=\"indent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert nach Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> ein <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>s<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mi>c<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> f\u00fcr welches die Ungleichung <math display=\"inline\"><mi>c<\/mi><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>c<\/mi><mi>s<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math> gilt. Dies zeigt die zweite charakterisierende Eigenschaft des Supremums und wir erhalten                                                                                                                                                                           <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mi>s<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z80ce696d015c\"> <a id=\"x1-64008r63\"><\/a> <span class=\"ecbx-1095\">Proposition 2.63 <\/span>(Supremum unter Summen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwei nicht-leere, von oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Teilmengen von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">von<\/span> <span class=\"ecti-1095\">oben beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir definieren <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi> <\/mrow> <\/msub> <\/math> und <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi> <\/mrow> <\/msub> <\/math> f\u00fcr alle <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> was <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> und <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math> impliziert. Da aber jedes Element von <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/math> von dieser Form ist, erhalten wir, dass <math display=\"inline\"><msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msub><\/math> eine obere Schranke von <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/math>                                                                                                                                                                           ist und dass <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><\/math> von oben beschr\u00e4nkt ist. <\/p><p class=\"indent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert nach Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> ein <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> mit <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> und ein <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac> <mrow> <mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> was wiederum <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math> impliziert. Dies zeigt die zweite charakterisierende Eigenschaft von <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> in Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> und wir erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>s<\/mi><\/mrow><mrow><mi>B<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <span>&nbsp;&nbsp;<\/span><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-64009r64\"><\/a> <h4 id=\"z293a3502e9ec\" class=\"subsectionHead\"><span class=\"titlemark\">2.5.3 <\/span> <a id=\"x1-650003\"><\/a>Uneigentliche Werte, Suprema und Infima<\/h4> <p class=\"noindent\">In diesem Abschnitt wollen wir die Begriffe \u201eSupremum\u201c und \u201eInfimum\u201c auf beliebige Teilmengen von <math display=\"inline\"><mi>\u211d<\/mi><\/math> erweitern (ohne die in Abschnitt&nbsp;<a href=\"..\/..\/chapter\/maximum-und-supremum#x1-640002\">2.5.2<\/a> getroffenen Annahmen). Dazu verwenden wir die Symbole <math display=\"inline\"><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> die keine reellen Zahlen darstellen. Wir definieren die <span class=\"ecbx-1095\">erweiterte Zahlengerade <\/span>(die auch <span class=\"ecbx-1095\">Zweipunktkompaktifizierung<\/span> von <math display=\"inline\"><mi>\u211d<\/mi><\/math> genannt wird) durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2294<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/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\">und stellen uns diese als die Zahlengerade <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=625&amp;url=Pictures\/Reelle_Zahlen\/Supremum\/uneig_sup.pdf\"><img decoding=\"async\" id=\"z50ef4f81a6cd\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Reelle_Zahlen\/Supremum\/uneig_sup.svg\" width=\"625\" \/><\/div>  <\/div> <p class=\"noindent\">vor. Hier haben wir den Punkt <math display=\"inline\"> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi><\/math> rechts von <math display=\"inline\"><mi>\u211d<\/mi><\/math> und den Punkt <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> links von <math display=\"inline\"><mi>\u211d<\/mi><\/math> zu der Gerade hinzugef\u00fcgt. Formaler formuliert: wir erweitern die Relation (Ordnung) <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> auf <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <\/math> gilt, aber keine weiteren <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math>-Relationen f\u00fcr die Symbole <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> erf\u00fcllt sind. Inbesondere schreiben wir auch <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z14e0617f0b27\"> <a id=\"x1-65001r64\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.64 <\/span>(Geometrie der Zweipunktkompaktifizierung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Abbildung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi>x<\/mi><\/mrow><\/mfrac> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/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\">bijektiv ist und die Ordnung erh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt. Das heisst, f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Erweitern Sie<\/span> <math display=\"inline\"><mi>\u03d5<\/mi><\/math> <span class=\"ecti-1095\">zu einer ordnungserhaltenden<\/span> <span class=\"ecti-1095\">Bijektion <\/span><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03d5<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-punc\">:<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren Sie damit das obige Bild der erweiterten Zahlengerade.<\/span> <\/p> <\/div> <p class=\"indent\">Das Maximum und das Minimum einer Teilmenge <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <\/math> ist nun wie in Abschnitt <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-630001\">2.5.1<\/a> definiert (falls es existiert). <\/p><p class=\"indent\">Falls <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> nicht von oben beschr\u00e4nkt ist, dann definieren wir <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>X<\/mi><\/math> leer ist, setzen wir <math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2205<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math> (da jedes <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> eine obere Schranke von <math display=\"inline\"><mi>\u2205<\/mi><\/math> darstellt). Analog definieren wir <math display=\"inline\"><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u2205<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> nicht von unten beschr\u00e4nkt ist. <\/p><p class=\"indent\">Folgende \u00dcbungen stellen nat\u00fcrliche Eigenschaften von Supremum und Infimum dar. Sie sollten mindestens eine dieser \u00dcbungen ausarbeiten. Betrachten Sie hierbei die Spezialf\u00e4lle, die zu einem uneigentlichen Supremum oder Infimum f\u00fchren, getrennt und gehen Sie anschliessend wie im Beweis von Proposition <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64008r63\">2.63<\/a> vor. <\/p><p class=\"indent\">Weiter definieren wir f\u00fcr die \u00dcbungen die \u201eRechenregeln\u201c                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mi>\u221e<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>\u221e<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi> <\/mtd> <mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><\/mtd><\/mtr> <\/mtable> <\/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\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>\u221e<\/mi><mo class=\"MathClass-bin\">\u22c5<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <\/mtd> <\/mtr><mtr><mtd class=\"array\" columnalign=\"center\"> <mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mi>\u221e<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <\/mtd> <\/mtr><mtr><mtd class=\"array\" columnalign=\"center\"> <mi>\u221e<\/mi><mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi> <\/mtd> <\/mtr><mtr><mtd class=\"array\" columnalign=\"center\"><mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/mtd><\/mtr> <\/mtable> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> wovon wir einen Teil verwenden werden. Die Ausdr\u00fccke <math display=\"inline\"><mi>\u221e<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u221e<\/mi><\/math> und <math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>\u221e<\/mi><\/math> oder \u00e4hnliche bleiben wohlgemerkt aber undefiniert. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb12bd7b9fd1c\"> <a id=\"x1-65002r65\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.65 <\/span>(Eigenschaften von Supremum und Infimum unter Vereinigung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">Teilmengen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>Y<\/mi> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Formulieren und beweisen Sie eine analoge Formel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Infimum von<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0fffa68e9585\"> <a id=\"x1-65003r66\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.66 <\/span>(Eigenschaften von Supremum und Infimum unter Summen und Produkten)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwei nicht-leere Teilmengen. Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><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\">und dass, falls <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Suchen Sie des Weiteren <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnliche Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Infimum.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z19c74f816cec\"> <a id=\"x1-65004r67\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.67.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">eine nicht-leere<\/span> <span class=\"ecti-1095\">Teilmenge von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><table id=\"zd79f609be996\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">Hierbei ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">das Bild von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">unter dem Absolutbetrag <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">(als Funktion von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <a id=\"x1-65005r65\"><\/a> <h4 id=\"z12efc5883900\" class=\"subsectionHead\"><span class=\"titlemark\">2.5.4 <\/span> <a id=\"x1-660004\"><\/a>Verwendung des Supremums und des Infimums<\/h4> <p class=\"noindent\">Das Supremum ist eine nat\u00fcrliche und notwendige Verallgemeinerung des Maximums einer Menge, da letzteres sogar f\u00fcr beschr\u00e4nkte Intervalle nicht existieren muss. Das Supremum kann aber auch hilfreich sein in Situationen, wo das Maximum existiert. Denn falls man beweisen will, dass ein Maximum existiert, dann hat man mit dem Supremum den richtigen Kandidaten und kann den Beweis mit der Existenz des Supremums beginnen. Auf die gleiche Weise ist das Infimum einer Menge eine Verallgemeinerung des Minimums. <\/p><p class=\"indent\">Es ist wichtig, dass Sie sich die charakterisierenden Eigenschaften des Supremums und Infimums einpr\u00e4gen, da diese Begriffe fundamentale Bausteine unserer zu entwickelnden Theorie sein werden. Zum Beispiel werden wir das Integral einer Funktion durch ein Supremum definieren (siehe Figur&nbsp;<a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4008r2\">1.2<\/a> und Kapitel&nbsp;<a href=\"..\/..\/part\/das-riemann-integral#x1-1060004\">4<\/a>).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-66001r62\"><\/a> <\/p> \n","protected":false},"author":1089,"menu_order":5,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-38","chapter","type-chapter","status-publish","hentry"],"part":33,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/38","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/38\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/33"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/38\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=38"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=38"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=38"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=38"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}