{"id":37,"date":"2021-12-15T09:53:00","date_gmt":"2021-12-15T09:53:00","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/intervalle-und-der-absolutbetrag\/"},"modified":"2021-12-15T09:53:00","modified_gmt":"2021-12-15T09:53:00","slug":"intervalle-und-der-absolutbetrag","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/intervalle-und-der-absolutbetrag\/","title":{"raw":"Intervalle und der Absolutbetrag","rendered":"Intervalle und der Absolutbetrag"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z0eef0a175ba2\" class=\"sectionHead\"><span class=\"titlemark\">2.4 <\/span> <a id=\"x1-580004\"><\/a>Intervalle und der Absolutbetrag<\/h3> <a id=\"x1-58001r57\"><\/a> <h4 id=\"z64a4dfac7449\" class=\"subsectionHead\"><span class=\"titlemark\">2.4.1 <\/span> <a id=\"x1-590001\"><\/a>Intervalle<\/h4> <p class=\"noindent\">Wie bereits erw\u00e4hnt, stellen wir <math display=\"inline\"><mi>\u211d<\/mi><\/math> als die Zahlengerade dar. In diesem Bild entsprechen folgende Teilmengen Strecken auf dieser Geraden, wobei wir vier M\u00f6glichkeiten haben, je nachdem, ob man die Endpunkte in der Teilmenge haben will oder nicht. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z90081b098612\"> <a id=\"x1-59001r41\"><\/a> <span class=\"ecbx-1095\">Definition 2.41 <\/span>(Intervalle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Seien <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann ist das <span class=\"ecbx-1095\">abgeschlossene Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/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\">das <span class=\"ecbx-1095\">offene Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/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\">das <span class=\"ecbx-1095\">(rechts) halboffene Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/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 das <span class=\"ecbx-1095\">(links) halboffene Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/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\">definiert. Wenn das Intervall nicht-leer ist, dann wird <math display=\"inline\"><mi>a<\/mi><\/math> der <span class=\"ecbx-1095\">linke Endpunkt<\/span>, <math display=\"inline\"><mi>b<\/mi><\/math> der <span class=\"ecbx-1095\">rechte<\/span> <span class=\"ecbx-1095\">Endpunkt<\/span>, und&nbsp;<math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math> die <span class=\"ecbx-1095\">L<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nge des Intervalls <\/span>genannt.                                                                                                                                                                           <\/p> <\/div> <p class=\"indent\">Wir m\u00f6chten an dieser Stelle anmerken, dass beispielsweise die Intervalle <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-punc\">,<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/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-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nicht-leer sind genau dann, wenn <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> und <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> nicht-leer ist genau dann, wenn <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Intervalle der Art <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> werden auch <span class=\"ecbx-1095\">endliche <\/span>oder <span class=\"ecbx-1095\">beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkte Intervalle <\/span>genannt, wenn wir sie von folgenden Intervallen unterscheiden wollen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z3cbd6cc92be6\"> <a id=\"x1-59002r42\"><\/a> <span class=\"ecbx-1095\">Definition 2.42 <\/span>(Unbeschr\u00e4nkte Intervalle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> definieren wir die <span class=\"ecbx-1095\">unbeschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkten abgeschlossenen Intervalle<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mi>a<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/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\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2264<\/mo><mi>b<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/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><\/mtable><\/math> <p class=\"noindent\">und die <span class=\"ecbx-1095\">unbeschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkten offenen Intervalle<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mi>a<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/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\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&lt;<\/mo><mi>b<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/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\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Statt runden Klammern werden manchmal auch umgedrehte eckige Klammern verwendet, um offene und halboffene Intervalle zu bezeichnen. Zum Beispiel findet man anstelle von <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> f\u00fcr <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> oft auch <math display=\"inline\"><mo class=\"MathClass-close\">]<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-open\">[<\/mo><\/math> in der Literatur. Wir werden hier stets runde Klammern verwenden. <\/p><p class=\"indent\">Der folgende Begriff wird f\u00fcr uns sp\u00e4ter sehr bedeutsam sein. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z65651f143f7f\"> <a id=\"x1-59003r43\"><\/a> <span class=\"ecbx-1095\">Definition 2.43 <\/span>(Umgebungen eines Punktes)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ein Menge, die ein offenes Intervall enth\u00e4lt, in dem <math display=\"inline\"><mi>x<\/mi><\/math> liegt, wird auch eine <span class=\"ecbx-1095\">Umgebung <\/span>von <math display=\"inline\"><mi>x<\/mi><\/math> genannt. F\u00fcr ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> wird das offene Intervall <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die <math display=\"inline\"><mi>\u03b4<\/mi><\/math><span class=\"ecbx-1095\">-Umgebung<\/span> von <math display=\"inline\"><mi>x<\/mi><\/math> genannt. <\/p> <\/div> <p class=\"indent\">Beispielsweise w\u00e4re also <math display=\"inline\"><mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u222a<\/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> eine Umgebung von <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls ein <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> in einer                                                                                                                                                                           <math display=\"inline\"><mi>\u03b4<\/mi><\/math>-Umgebung eines Punktes <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> liegt f\u00fcr ein \u201e kleines\u201c <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so sagt man auch, dass <math display=\"inline\"><mi>y<\/mi><\/math> \u201e<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-nahe\u201c an <math display=\"inline\"><mi>x<\/mi><\/math> ist. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z72d8c3169a9c\"> <a id=\"x1-59004r44\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.44 <\/span>(Verhalten von Intervallen unter Durchschnitt und Vereinigung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass ein endlicher Schnitt <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u22c2<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">von Intervallen <\/span><math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">wieder ein Intervall ist (wobei die leere Menge auch als ein Intervall zugelassen ist).<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie die Endpunkte eines nicht-leeren Durchschnitts mittels der Endpunkte der<\/span> <span class=\"ecti-1095\">urspr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nglichen Intervalle beschreiben?<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wann ist eine Vereinigung von zwei Intervallen wieder ein Intervall? Was geschieht<\/span> <span class=\"ecti-1095\">in diesem Fall, wenn man zwei Intervalle des selben Typs (offen, abgeschlossen, links<\/span> <span class=\"ecti-1095\">halboffen, rechts halboffen) vereinigt?<\/span><\/dd><\/dl> <\/div> <a id=\"x1-59007r59\"><\/a> <h4 id=\"ze379aff2591e\" class=\"subsectionHead\"><span class=\"titlemark\">2.4.2 <\/span> <a id=\"x1-600002\"><\/a>Der Absolutbetrag auf den reellen Zahlen<\/h4> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z1182ad108ecc\"> <a id=\"x1-60001r45\"><\/a> <span class=\"ecbx-1095\">Definition 2.45.<\/span> <\/h4> <p class=\"indent\">Der <span class=\"ecbx-1095\">Absolutbetrag <\/span>ist die Funktion                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mo class=\"MathClass-bin\">\u22c5<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><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><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/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=\"center\"> <mi>x<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"><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=\"center\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mtd> <mtd class=\"array\" columnalign=\"center\"><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><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 betrachten zuerst einige Konsequenzen dieser Definition. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z28127916215e\"> <span class=\"ecbx-1095\">Folgerungen.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">genau dann, wenn <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dies folgt aus der Trichotomie von reellen Zahlen: F\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> und f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> folgt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Absolutbetrag ist multiplikativ: <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> (\u00dcberpr\u00fcfen Sie dies in den insgesamt vier F\u00e4llen, je nachdem, ob <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>y<\/mi><\/math> negativ sind oder nicht.) <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(d)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Dies folgt aus (c) wegen <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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>                                                                                                                                                                           <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(e)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent zu <\/span><span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Denn angenommen <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> dann gilt <span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> dann ist <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> und damit wiederum <span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr die Umkehrung bemerken wir, dass <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> auch <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> und somit in jedem Fall <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> impliziert. <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(f)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Analog ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/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\">die strikte Ungleichung <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent zu <\/span><span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(g)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Dreiecksungleichung) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/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=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/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\">Diese Ungleichung wird auch die <\/span><span class=\"ecbi-1095\">Dreiecksungleichung <\/span><span class=\"ecti-1095\">genannt. <\/span>Sie folgt, in dem wir <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> und <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> wie in (e) addieren und anschliessend auf <\/p><table id=\"zc06577592303\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"noindent\">wiederum Eigenschaft (e) anwenden. <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(h)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(umgekehrte Dreiecksungleichung) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/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=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Denn die Dreiecksungleichung in (g) zeigt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was zu <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fchrt. Durch Vertauschen von <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/math> erhalten wir <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> Also ist nach Eigenschaft (e) <math display=\"inline\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> wie gew\u00fcnscht.<\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfd3a2aa246be\"> <a id=\"x1-60010r46\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.46.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                                                                                                 welche<\/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 Gleichheit in der Dreiecksungleichung oder der umgekehrten Dreiecksungleichung?<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"><mi class=\"qopname\">sgn<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das <span class=\"ecbx-1095\">Vorzeichen<\/span> (oder <span class=\"ecbx-1095\">Signum<\/span>) von <math display=\"inline\"><mi>x<\/mi><\/math> ist, welches durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><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> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/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\">definiert ist. Das Vorzeichen einer Zahlen ist also genau dann <math display=\"inline\"><mn>1<\/mn><\/math> (respektive <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math>), wenn die Zahl positiv (respektive negativ) ist. Die Zahl <math display=\"inline\"><mn>0<\/mn><\/math> ist weder positiv noch negativ und deswegen weist man ihr das \u201eVorzeichen Null\u201c zu. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf47f100d0aa8\"> <a id=\"x1-60011r47\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.47 <\/span>(Absolutbetrag und Quadratwurzel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die Gleichungen <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Die Wurzelfunktion wurde in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\"><span class=\"ecti-1095\">2.11<\/span><\/a> <span class=\"ecti-1095\">eingef<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wir bemerken noch, dass f\u00fcr <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die <math display=\"inline\"><mi>\u03b4<\/mi><\/math>-Umgebung von <math display=\"inline\"><mi>x<\/mi><\/math> (siehe Definition&nbsp;<a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-59003r43\">2.43<\/a>) durch <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gegeben ist. Wir werden <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> als den <span class=\"ecbx-1095\">Abstand <\/span>von <math display=\"inline\"><mi>x<\/mi><\/math> zu <math display=\"inline\"><mi>y<\/mi><\/math> interpretieren. Im Sinne des Wortes \u201eAbstand\u201c kann man ein paar der obigen Folgerungen neu intuitiver ausdr\u00fccken. Zum Beispiel besagt (b), dass f\u00fcr <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die Gleichheit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> erf\u00fcllt ist, was also bedeutet, dass der Abstand von <math display=\"inline\"><mi>x<\/mi><\/math> zu <math display=\"inline\"><mi>y<\/mi><\/math> dem Abstand von <math display=\"inline\"><mi>y<\/mi><\/math> zu <math display=\"inline\"><mi>x<\/mi><\/math> gleich ist (wie man sich w\u00fcnschen k\u00f6nnte). Des Weiteren werden Umgebungen einer reellen Zahl <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> auch <span class=\"ecbx-1095\">Nachbarschaften <\/span>von <math display=\"inline\"><mi>x<\/mi><\/math> genannt. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zf6e2d44508c0\"> <a id=\"x1-60012r48\"><\/a> <span class=\"ecbx-1095\">Definition 2.48 <\/span>(Offene und abgeschlossene Teilmengen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Teilmenge <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> heisst <span class=\"ecbx-1095\">offen <\/span>(in <math display=\"inline\"><mi>\u211d<\/mi><\/math>), wenn f\u00fcr jedes <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>U<\/mi><\/math>                                                                                                                                                                           ein <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Eine Teilmenge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> heisst <span class=\"ecbx-1095\">abgeschlossen <\/span>(in <math display=\"inline\"><mi>\u211d<\/mi><\/math>), wenn ihr Komplement <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>A<\/mi><\/math> offen ist. <\/p> <\/div> <p class=\"indent\">Intuitiv ausgedr\u00fcckt ist eine Teilmenge offen, wenn f\u00fcr jeden Punkt <math display=\"inline\"><mi>x<\/mi><\/math> in der Menge alle Punkte, die nahe genug an <math display=\"inline\"><mi>x<\/mi><\/math> sind, wieder in der Menge liegen. Wir kennen bereits Beispiele von offenen Mengen: <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zae438710e839\"> <a id=\"x1-60013r49\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.49 <\/span>(Offene Intervalle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                   Sie,                   dass                   eine                   Teilmenge<\/span> <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">genau dann          offen          ist,          wenn          f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r          jeden          Punkt<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>U<\/mi><\/math> <span class=\"ecti-1095\">ein                                              offenes                                              Intervall<\/span> <math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><\/math> <span class=\"ecti-1095\">existiert. Schliessen Sie, dass die offenen (respektive abgeschlossenen) Intervalle auch im Sinne<\/span> <span class=\"ecti-1095\">der obigen Definition offen (respektive abgeschlossen) sind.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5d5bba64fc75\"> <a id=\"x1-60014r50\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.50.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Entscheiden Sie bei den folgenden Teilmengen von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">jeweils, ob sie offen, abgeschlossen oder weder noch sind.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Teilmengen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2205<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Teilmengen <\/span><span class=\"maperiod\"><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><\/span><span class=\"period\">,<\/span> <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> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u222a<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <a id=\"x1-60015r60\"><\/a> <h4 id=\"z72749fad3507\" class=\"subsectionHead\"><span class=\"titlemark\">2.4.3 <\/span> <a id=\"x1-610003\"><\/a>Der Absolutbetrag auf den komplexen Zahlen<\/h4> <p class=\"noindent\">Wir m\u00f6chten nun den Absolutbetrag auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> so definieren, so dass dieser m\u00f6glichst viele Eigenschaften des Absolutbetrags auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> hat und mit diesem kompatibel ist. Wir verwenden dazu die Wurzelfunktion, die in \u00dcbung <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\">2.11<\/a> eingef\u00fchrt wurde. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z2d60528c374c\"> <a id=\"x1-61001r51\"><\/a> <span class=\"ecbx-1095\">Definition 2.51.<\/span> <\/h4> <p class=\"indent\">Der <span class=\"ecbx-1095\">Absolutbetrag <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecbx-1095\">auf <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> ist gegeben durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">An dieser Stelle bemerken wir, dass f\u00fcr <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> die Summe der Quadrate <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> gerade gleich <math display=\"inline\"><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> ist, denn <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Somit gilt f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt><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\">Des Weiteren m\u00f6chten wir anmerken, dass f\u00fcr ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> der zu Beginn von Abschnitt <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-600002\">2.4.2<\/a> definierte Absolutbetrag <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> und der Absolutbetrag von <math display=\"inline\"><mi>x<\/mi><\/math> als Element von <math display=\"inline\"><mi>\u2102<\/mi><\/math> \u00fcbereinstimmen, da <math display=\"inline\"><msqrt><mrow><mi>x<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math> (vergleiche \u00dcbung <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-60011r47\">2.47<\/a>). Insbesondere ist die neu eingef\u00fchrte Notation nicht widerspr\u00fcchlich und wir haben den Absolutbetrag von <math display=\"inline\"><mi>\u211d<\/mi><\/math> auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> erweitert. <\/p><p class=\"indent\">Wir fassen nun einige Eigenschaften des Absolutbetrags auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> zusammen: <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z15a706567ace\"> <span class=\"ecbx-1095\">Eigenschaften des Absolutbetrags auf <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math><span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Definitheit) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">genau dann, wenn <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Multiplikativit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Dreiecksungleichung) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Umgekehrte Dreiecksungleichung) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Genauso wie auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> wollen wir mit Hilfe des Absolutbetrags den <span class=\"ecbx-1095\">Abstand <\/span>zweier Punkte <math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> als die nicht-negative Zahl <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> auffassen. Wir bemerken noch, dass Definition&nbsp;<a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-61001r51\">2.51<\/a> dem Satz von Pythagoras (siehe die entsprechende \u00dcbung in Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-390006\">1.9.6<\/a>) entspricht. Doch haben wir dies als Definition des Absolutbetrages von <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <\/math> gew\u00e4hlt, womit es (abgesehen von obigen Eigenschaften) nichts zu beweisen gibt. <\/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\">Zur Definitheit: Per Definition der Wurzel gilt f\u00fcr ein <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> dass <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Des Weiteren gilt <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> wegen der Injektivit\u00e4t der Wurzelfunktion genau dann, wenn <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> In Lemma <a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56008r37\">2.37<\/a> wurde jedoch gezeigt, dass <math display=\"inline\"><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> genau dann Null ist, wenn <math display=\"inline\"><mi>z<\/mi><\/math> selbst Null ist. Also folgt die Definitheit des Absolutbetrags. <\/p><p class=\"indent\">F\u00fcr die Multiplikativit\u00e4t verwenden wir die Eigenschaften der Konjugation aus Lemma&nbsp;<a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56008r37\">2.37<\/a> und die Multiplikativit\u00e4t der Wurzel (siehe \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\">2.11<\/a>(vi)). Seien <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><mi>w<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>z<\/mi><mi>w<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mi>w<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt><msqrt><mrow><mi>w<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>w<\/mi><\/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\">was zu zeigen war.                                                                                                                                                                           <\/p><p class=\"indent\">F\u00fcr die Dreiecksungleichung betrachten wir <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da die Wurzelfunktion Ungleichungen zwischen positive Zahlen erh\u00e4lt (siehe \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\">2.11<\/a>(iv)), reicht es die Ungleichung <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> zu zeigen. Wir berechnen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><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\">Wie wir sehen werden, reicht es aus die Ungleichung <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> zu zeigen, die auch als Cauchy-Schwarz-Ungleichung auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> bekannt ist. Tats\u00e4chlich gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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\">und daher auch&nbsp;<span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> Zusammen ergibt sich <\/p><table id=\"zf63c8c8edfb0\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/math><\/td><\/tr><\/table> <p class=\"indent\">Die umgekehrte Dreiecksungleichung folgt ebenso wie im reellen Fall direkt aus der Dreiecksungleichung. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wie vorhin l\u00e4sst sich mit Hilfe des Absolutbetrags ein Begriff von Offen- und Abgeschlossenheit einf\u00fchren. F\u00fcr die Definition von offenen Mengen in <math display=\"inline\"><mi>\u211d<\/mi><\/math> wurden die symmetrisch um einen zuvor fixierten Punkt liegenden offenen Intervalle verwendet. In Analogie dazu definieren wir folgende Teilmengen von <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z89a31db1743d\"> <a id=\"x1-61006r52\"><\/a> <span class=\"ecbx-1095\">Definition 2.52 <\/span>(Offene B\u00e4lle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Der <span class=\"ecbx-1095\">offene Ball <\/span>mit Radius <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> um einen Punkt <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> ist die Menge <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>r<\/mi><\/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> <\/div> <p class=\"indent\">Der offene Ball <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> zu <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> besteht also gerade aus jenen Punkten, die Abstand (strikt) kleiner <math display=\"inline\"><mi>r<\/mi><\/math> von <math display=\"inline\"><mi>z<\/mi><\/math> haben. Offene B\u00e4lle in <math display=\"inline\"><mi>\u2102<\/mi><\/math> und offene Intervalle in <math display=\"inline\"><mi>\u211d<\/mi><\/math> sind in folgendem Sinne kompatibel: Ist <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so ist der Schnitt des offenen Balles <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mi>\u211d<\/mi><\/math> gerade das offene, symmetrisch um <math display=\"inline\"><mi>x<\/mi><\/math> liegende Intervall <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>r<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> (wieso?). <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=339&amp;url=Pictures\/Reelle_Zahlen\/komplexe_Ebene\/balls.pdf\"><img id=\"z9229334e039b\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Reelle_Zahlen\/komplexe_Ebene\/balls.svg\" width=\"339\"><\/div>  <\/div> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"zf48355508708\"> <a id=\"x1-61007r53\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.53 <\/span>(Durchschnitt von offenen B\u00e4llen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie folgende Eigenschaft von B<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llen: Seien<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jeden<\/span> <span class=\"ecti-1095\">Punkt <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">existiert<\/span> <span class=\"ecti-1095\">ein Radius <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Illustrieren Sie Ihre Wahl des Radius <\/span><math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">in einem Bild.<\/span> <\/p> <\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zcc9456698ff5\"> <a id=\"x1-61008r54\"><\/a> <span class=\"ecbx-1095\">Definition 2.54 <\/span>(Offene und abgeschlossene Teilmengen von <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Teilmenge <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> heisst <span class=\"ecbx-1095\">offen <\/span>(in <math display=\"inline\"><mi>\u2102<\/mi><\/math>), wenn zu jedem Punkt in <math display=\"inline\"><mi>U<\/mi><\/math> ein offener Ball um diesen Punkt existiert, der in <math display=\"inline\"><mi>U<\/mi><\/math> enthalten ist. Formaler: F\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>U<\/mi><\/math> existiert ein Radius <span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><\/math><\/span><span class=\"period\">.<\/span> Eine Teilmenge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> heisst <span class=\"ecbx-1095\">abgeschlossen <\/span>(in <math display=\"inline\"><mi>\u2102<\/mi><\/math>), falls ihr Komplement <math display=\"inline\"><mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>A<\/mi><\/math> offen ist. <\/p> <\/div> <p class=\"indent\">Nach \u00dcbung <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-61007r53\">2.53<\/a> sind beispielsweise alle B\u00e4lle offen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1c8b1f6e22c7\"> <a id=\"x1-61009r55\"><\/a> <span class=\"ecbx-1095\">Applet 2.55 <\/span>(Offener Ball)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"495px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/t3nn8gv3\/width\/688\/height\/495\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir            sehen,            dass            es            f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r            jeden            Punkt<\/span> <math display=\"inline\"><mi>w<\/mi><\/math> <span class=\"ecti-1095\">in                               dem                               offenen                               Ball<\/span> <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">um<\/span> <math display=\"inline\"><mi>z<\/mi><\/math> <span class=\"ecti-1095\">mit                                                                                                      Radius<\/span> <math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">wieder                                              einen                                              Radius<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gibt,              so              dass              der              offene              Ball              um<\/span> <math display=\"inline\"><mi>w<\/mi><\/math> <span class=\"ecti-1095\">mit                                                                                                      Radius<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">ganz                                                                                                          in<\/span> <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">enthalten ist.<\/span> <\/p> <\/div> <p class=\"indent\">Es gibt, abgesehen von den offenen B\u00e4llen, noch viele weitere, offene Teilmengen von <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Beispielsweise ist jede Vereinigung von offenen Teilmengen offen. Zum Studium der offenen Mengen und damit verwandten Begriffen werden wir in deutlicher gr\u00f6sserer Allgemeinheit im zweiten Semester zur\u00fcckkehren. Insbesondere wollen wir uns hier noch nicht auf eine ausf\u00fchrliche Diskussion einlassen.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-61010r58\"><\/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=\"z0eef0a175ba2\" class=\"sectionHead\"><span class=\"titlemark\">2.4 <\/span> <a id=\"x1-580004\"><\/a>Intervalle und der Absolutbetrag<\/h3> <a id=\"x1-58001r57\"><\/a> <h4 id=\"z64a4dfac7449\" class=\"subsectionHead\"><span class=\"titlemark\">2.4.1 <\/span> <a id=\"x1-590001\"><\/a>Intervalle<\/h4> <p class=\"noindent\">Wie bereits erw\u00e4hnt, stellen wir <math display=\"inline\"><mi>\u211d<\/mi><\/math> als die Zahlengerade dar. In diesem Bild entsprechen folgende Teilmengen Strecken auf dieser Geraden, wobei wir vier M\u00f6glichkeiten haben, je nachdem, ob man die Endpunkte in der Teilmenge haben will oder nicht. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z90081b098612\"> <a id=\"x1-59001r41\"><\/a> <span class=\"ecbx-1095\">Definition 2.41 <\/span>(Intervalle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Seien <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann ist das <span class=\"ecbx-1095\">abgeschlossene Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/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\">das <span class=\"ecbx-1095\">offene Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/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\">das <span class=\"ecbx-1095\">(rechts) halboffene Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/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 das <span class=\"ecbx-1095\">(links) halboffene Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/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\">definiert. Wenn das Intervall nicht-leer ist, dann wird <math display=\"inline\"><mi>a<\/mi><\/math> der <span class=\"ecbx-1095\">linke Endpunkt<\/span>, <math display=\"inline\"><mi>b<\/mi><\/math> der <span class=\"ecbx-1095\">rechte<\/span> <span class=\"ecbx-1095\">Endpunkt<\/span>, und&nbsp;<math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math> die <span class=\"ecbx-1095\">L<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nge des Intervalls <\/span>genannt.                                                                                                                                                                           <\/p> <\/div> <p class=\"indent\">Wir m\u00f6chten an dieser Stelle anmerken, dass beispielsweise die Intervalle <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-punc\">,<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/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-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nicht-leer sind genau dann, wenn <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> und <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> nicht-leer ist genau dann, wenn <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Intervalle der Art <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> werden auch <span class=\"ecbx-1095\">endliche <\/span>oder <span class=\"ecbx-1095\">beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkte Intervalle <\/span>genannt, wenn wir sie von folgenden Intervallen unterscheiden wollen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3cbd6cc92be6\"> <a id=\"x1-59002r42\"><\/a> <span class=\"ecbx-1095\">Definition 2.42 <\/span>(Unbeschr\u00e4nkte Intervalle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> definieren wir die <span class=\"ecbx-1095\">unbeschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkten abgeschlossenen Intervalle<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mi>a<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/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\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2264<\/mo><mi>b<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/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><\/mtable><\/math> <p class=\"noindent\">und die <span class=\"ecbx-1095\">unbeschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkten offenen Intervalle<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mi>a<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/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\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&lt;<\/mo><mi>b<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/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\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Statt runden Klammern werden manchmal auch umgedrehte eckige Klammern verwendet, um offene und halboffene Intervalle zu bezeichnen. Zum Beispiel findet man anstelle von <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> f\u00fcr <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> oft auch <math display=\"inline\"><mo class=\"MathClass-close\">]<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-open\">[<\/mo><\/math> in der Literatur. Wir werden hier stets runde Klammern verwenden. <\/p><p class=\"indent\">Der folgende Begriff wird f\u00fcr uns sp\u00e4ter sehr bedeutsam sein. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z65651f143f7f\"> <a id=\"x1-59003r43\"><\/a> <span class=\"ecbx-1095\">Definition 2.43 <\/span>(Umgebungen eines Punktes)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ein Menge, die ein offenes Intervall enth\u00e4lt, in dem <math display=\"inline\"><mi>x<\/mi><\/math> liegt, wird auch eine <span class=\"ecbx-1095\">Umgebung <\/span>von <math display=\"inline\"><mi>x<\/mi><\/math> genannt. F\u00fcr ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> wird das offene Intervall <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die <math display=\"inline\"><mi>\u03b4<\/mi><\/math><span class=\"ecbx-1095\">-Umgebung<\/span> von <math display=\"inline\"><mi>x<\/mi><\/math> genannt. <\/p> <\/div> <p class=\"indent\">Beispielsweise w\u00e4re also <math display=\"inline\"><mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u222a<\/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> eine Umgebung von <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls ein <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> in einer                                                                                                                                                                           <math display=\"inline\"><mi>\u03b4<\/mi><\/math>-Umgebung eines Punktes <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> liegt f\u00fcr ein \u201e kleines\u201c <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so sagt man auch, dass <math display=\"inline\"><mi>y<\/mi><\/math> \u201e<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-nahe\u201c an <math display=\"inline\"><mi>x<\/mi><\/math> ist. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z72d8c3169a9c\"> <a id=\"x1-59004r44\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.44 <\/span>(Verhalten von Intervallen unter Durchschnitt und Vereinigung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass ein endlicher Schnitt <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u22c2<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">von Intervallen <\/span><math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">wieder ein Intervall ist (wobei die leere Menge auch als ein Intervall zugelassen ist).<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie die Endpunkte eines nicht-leeren Durchschnitts mittels der Endpunkte der<\/span> <span class=\"ecti-1095\">urspr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nglichen Intervalle beschreiben?<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wann ist eine Vereinigung von zwei Intervallen wieder ein Intervall? Was geschieht<\/span> <span class=\"ecti-1095\">in diesem Fall, wenn man zwei Intervalle des selben Typs (offen, abgeschlossen, links<\/span> <span class=\"ecti-1095\">halboffen, rechts halboffen) vereinigt?<\/span><\/dd><\/dl> <\/div> <a id=\"x1-59007r59\"><\/a> <h4 id=\"ze379aff2591e\" class=\"subsectionHead\"><span class=\"titlemark\">2.4.2 <\/span> <a id=\"x1-600002\"><\/a>Der Absolutbetrag auf den reellen Zahlen<\/h4> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1182ad108ecc\"> <a id=\"x1-60001r45\"><\/a> <span class=\"ecbx-1095\">Definition 2.45.<\/span> <\/h4> <p class=\"indent\">Der <span class=\"ecbx-1095\">Absolutbetrag <\/span>ist die Funktion                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mo class=\"MathClass-bin\">\u22c5<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><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><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/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=\"center\"> <mi>x<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"><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=\"center\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mtd> <mtd class=\"array\" columnalign=\"center\"><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><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 betrachten zuerst einige Konsequenzen dieser Definition. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z28127916215e\"> <span class=\"ecbx-1095\">Folgerungen.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">genau dann, wenn <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dies folgt aus der Trichotomie von reellen Zahlen: F\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> und f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> folgt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Absolutbetrag ist multiplikativ: <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> (\u00dcberpr\u00fcfen Sie dies in den insgesamt vier F\u00e4llen, je nachdem, ob <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>y<\/mi><\/math> negativ sind oder nicht.) <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(d)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Dies folgt aus (c) wegen <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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>                                                                                                                                                                           <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(e)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent zu <\/span><span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Denn angenommen <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> dann gilt <span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> dann ist <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> und damit wiederum <span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr die Umkehrung bemerken wir, dass <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> auch <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> und somit in jedem Fall <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> impliziert. <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(f)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Analog ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/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\">die strikte Ungleichung <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent zu <\/span><span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(g)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Dreiecksungleichung) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/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=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/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\">Diese Ungleichung wird auch die <\/span><span class=\"ecbi-1095\">Dreiecksungleichung <\/span><span class=\"ecti-1095\">genannt. <\/span>Sie folgt, in dem wir <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> und <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> wie in (e) addieren und anschliessend auf <\/p><table id=\"zc06577592303\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"noindent\">wiederum Eigenschaft (e) anwenden. <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(h)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(umgekehrte Dreiecksungleichung) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/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=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Denn die Dreiecksungleichung in (g) zeigt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was zu <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fchrt. Durch Vertauschen von <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/math> erhalten wir <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> Also ist nach Eigenschaft (e) <math display=\"inline\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> wie gew\u00fcnscht.<\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfd3a2aa246be\"> <a id=\"x1-60010r46\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.46.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                                                                                                 welche<\/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 Gleichheit in der Dreiecksungleichung oder der umgekehrten Dreiecksungleichung?<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"><mi class=\"qopname\">sgn<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das <span class=\"ecbx-1095\">Vorzeichen<\/span> (oder <span class=\"ecbx-1095\">Signum<\/span>) von <math display=\"inline\"><mi>x<\/mi><\/math> ist, welches durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><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> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/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\">definiert ist. Das Vorzeichen einer Zahlen ist also genau dann <math display=\"inline\"><mn>1<\/mn><\/math> (respektive <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math>), wenn die Zahl positiv (respektive negativ) ist. Die Zahl <math display=\"inline\"><mn>0<\/mn><\/math> ist weder positiv noch negativ und deswegen weist man ihr das \u201eVorzeichen Null\u201c zu. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf47f100d0aa8\"> <a id=\"x1-60011r47\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.47 <\/span>(Absolutbetrag und Quadratwurzel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die Gleichungen <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Die Wurzelfunktion wurde in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\"><span class=\"ecti-1095\">2.11<\/span><\/a> <span class=\"ecti-1095\">eingef<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wir bemerken noch, dass f\u00fcr <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die <math display=\"inline\"><mi>\u03b4<\/mi><\/math>-Umgebung von <math display=\"inline\"><mi>x<\/mi><\/math> (siehe Definition&nbsp;<a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-59003r43\">2.43<\/a>) durch <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gegeben ist. Wir werden <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> als den <span class=\"ecbx-1095\">Abstand <\/span>von <math display=\"inline\"><mi>x<\/mi><\/math> zu <math display=\"inline\"><mi>y<\/mi><\/math> interpretieren. Im Sinne des Wortes \u201eAbstand\u201c kann man ein paar der obigen Folgerungen neu intuitiver ausdr\u00fccken. Zum Beispiel besagt (b), dass f\u00fcr <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die Gleichheit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> erf\u00fcllt ist, was also bedeutet, dass der Abstand von <math display=\"inline\"><mi>x<\/mi><\/math> zu <math display=\"inline\"><mi>y<\/mi><\/math> dem Abstand von <math display=\"inline\"><mi>y<\/mi><\/math> zu <math display=\"inline\"><mi>x<\/mi><\/math> gleich ist (wie man sich w\u00fcnschen k\u00f6nnte). Des Weiteren werden Umgebungen einer reellen Zahl <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> auch <span class=\"ecbx-1095\">Nachbarschaften <\/span>von <math display=\"inline\"><mi>x<\/mi><\/math> genannt. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf6e2d44508c0\"> <a id=\"x1-60012r48\"><\/a> <span class=\"ecbx-1095\">Definition 2.48 <\/span>(Offene und abgeschlossene Teilmengen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Teilmenge <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> heisst <span class=\"ecbx-1095\">offen <\/span>(in <math display=\"inline\"><mi>\u211d<\/mi><\/math>), wenn f\u00fcr jedes <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>U<\/mi><\/math>                                                                                                                                                                           ein <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Eine Teilmenge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> heisst <span class=\"ecbx-1095\">abgeschlossen <\/span>(in <math display=\"inline\"><mi>\u211d<\/mi><\/math>), wenn ihr Komplement <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>A<\/mi><\/math> offen ist. <\/p> <\/div> <p class=\"indent\">Intuitiv ausgedr\u00fcckt ist eine Teilmenge offen, wenn f\u00fcr jeden Punkt <math display=\"inline\"><mi>x<\/mi><\/math> in der Menge alle Punkte, die nahe genug an <math display=\"inline\"><mi>x<\/mi><\/math> sind, wieder in der Menge liegen. Wir kennen bereits Beispiele von offenen Mengen: <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zae438710e839\"> <a id=\"x1-60013r49\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.49 <\/span>(Offene Intervalle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                   Sie,                   dass                   eine                   Teilmenge<\/span> <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">genau dann          offen          ist,          wenn          f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r          jeden          Punkt<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>U<\/mi><\/math> <span class=\"ecti-1095\">ein                                              offenes                                              Intervall<\/span> <math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><\/math> <span class=\"ecti-1095\">existiert. Schliessen Sie, dass die offenen (respektive abgeschlossenen) Intervalle auch im Sinne<\/span> <span class=\"ecti-1095\">der obigen Definition offen (respektive abgeschlossen) sind.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5d5bba64fc75\"> <a id=\"x1-60014r50\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.50.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Entscheiden Sie bei den folgenden Teilmengen von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">jeweils, ob sie offen, abgeschlossen oder weder noch sind.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Teilmengen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2205<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Teilmengen <\/span><span class=\"maperiod\"><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><\/span><span class=\"period\">,<\/span> <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> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u222a<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <a id=\"x1-60015r60\"><\/a> <h4 id=\"z72749fad3507\" class=\"subsectionHead\"><span class=\"titlemark\">2.4.3 <\/span> <a id=\"x1-610003\"><\/a>Der Absolutbetrag auf den komplexen Zahlen<\/h4> <p class=\"noindent\">Wir m\u00f6chten nun den Absolutbetrag auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> so definieren, so dass dieser m\u00f6glichst viele Eigenschaften des Absolutbetrags auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> hat und mit diesem kompatibel ist. Wir verwenden dazu die Wurzelfunktion, die in \u00dcbung <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\">2.11<\/a> eingef\u00fchrt wurde. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2d60528c374c\"> <a id=\"x1-61001r51\"><\/a> <span class=\"ecbx-1095\">Definition 2.51.<\/span> <\/h4> <p class=\"indent\">Der <span class=\"ecbx-1095\">Absolutbetrag <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecbx-1095\">auf <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> ist gegeben durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">An dieser Stelle bemerken wir, dass f\u00fcr <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> die Summe der Quadrate <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> gerade gleich <math display=\"inline\"><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> ist, denn <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Somit gilt f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt><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\">Des Weiteren m\u00f6chten wir anmerken, dass f\u00fcr ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> der zu Beginn von Abschnitt <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-600002\">2.4.2<\/a> definierte Absolutbetrag <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> und der Absolutbetrag von <math display=\"inline\"><mi>x<\/mi><\/math> als Element von <math display=\"inline\"><mi>\u2102<\/mi><\/math> \u00fcbereinstimmen, da <math display=\"inline\"><msqrt><mrow><mi>x<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>x<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/math> (vergleiche \u00dcbung <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-60011r47\">2.47<\/a>). Insbesondere ist die neu eingef\u00fchrte Notation nicht widerspr\u00fcchlich und wir haben den Absolutbetrag von <math display=\"inline\"><mi>\u211d<\/mi><\/math> auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> erweitert. <\/p><p class=\"indent\">Wir fassen nun einige Eigenschaften des Absolutbetrags auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> zusammen: <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z15a706567ace\"> <span class=\"ecbx-1095\">Eigenschaften des Absolutbetrags auf <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math><span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Definitheit) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">genau dann, wenn <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Multiplikativit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Dreiecksungleichung) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Umgekehrte Dreiecksungleichung) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Genauso wie auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> wollen wir mit Hilfe des Absolutbetrags den <span class=\"ecbx-1095\">Abstand <\/span>zweier Punkte <math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> als die nicht-negative Zahl <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> auffassen. Wir bemerken noch, dass Definition&nbsp;<a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-61001r51\">2.51<\/a> dem Satz von Pythagoras (siehe die entsprechende \u00dcbung in Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-390006\">1.9.6<\/a>) entspricht. Doch haben wir dies als Definition des Absolutbetrages von <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <\/math> gew\u00e4hlt, womit es (abgesehen von obigen Eigenschaften) nichts zu beweisen gibt. <\/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\">Zur Definitheit: Per Definition der Wurzel gilt f\u00fcr ein <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> dass <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Des Weiteren gilt <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> wegen der Injektivit\u00e4t der Wurzelfunktion genau dann, wenn <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> In Lemma <a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56008r37\">2.37<\/a> wurde jedoch gezeigt, dass <math display=\"inline\"><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> genau dann Null ist, wenn <math display=\"inline\"><mi>z<\/mi><\/math> selbst Null ist. Also folgt die Definitheit des Absolutbetrags. <\/p><p class=\"indent\">F\u00fcr die Multiplikativit\u00e4t verwenden wir die Eigenschaften der Konjugation aus Lemma&nbsp;<a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56008r37\">2.37<\/a> und die Multiplikativit\u00e4t der Wurzel (siehe \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\">2.11<\/a>(vi)). Seien <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><mi>w<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>z<\/mi><mi>w<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mi>w<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>z<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt><msqrt><mrow><mi>w<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>w<\/mi><\/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\">was zu zeigen war.                                                                                                                                                                           <\/p><p class=\"indent\">F\u00fcr die Dreiecksungleichung betrachten wir <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da die Wurzelfunktion Ungleichungen zwischen positive Zahlen erh\u00e4lt (siehe \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\">2.11<\/a>(iv)), reicht es die Ungleichung <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> zu zeigen. Wir berechnen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><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\">Wie wir sehen werden, reicht es aus die Ungleichung <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> zu zeigen, die auch als Cauchy-Schwarz-Ungleichung auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> bekannt ist. Tats\u00e4chlich gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>x<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>y<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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\">und daher auch&nbsp;<span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> Zusammen ergibt sich <\/p><table id=\"zf63c8c8edfb0\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/math><\/td><\/tr><\/table> <p class=\"indent\">Die umgekehrte Dreiecksungleichung folgt ebenso wie im reellen Fall direkt aus der Dreiecksungleichung. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wie vorhin l\u00e4sst sich mit Hilfe des Absolutbetrags ein Begriff von Offen- und Abgeschlossenheit einf\u00fchren. F\u00fcr die Definition von offenen Mengen in <math display=\"inline\"><mi>\u211d<\/mi><\/math> wurden die symmetrisch um einen zuvor fixierten Punkt liegenden offenen Intervalle verwendet. In Analogie dazu definieren wir folgende Teilmengen von <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z89a31db1743d\"> <a id=\"x1-61006r52\"><\/a> <span class=\"ecbx-1095\">Definition 2.52 <\/span>(Offene B\u00e4lle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Der <span class=\"ecbx-1095\">offene Ball <\/span>mit Radius <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> um einen Punkt <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> ist die Menge <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>r<\/mi><\/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> <\/div> <p class=\"indent\">Der offene Ball <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> zu <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> besteht also gerade aus jenen Punkten, die Abstand (strikt) kleiner <math display=\"inline\"><mi>r<\/mi><\/math> von <math display=\"inline\"><mi>z<\/mi><\/math> haben. Offene B\u00e4lle in <math display=\"inline\"><mi>\u2102<\/mi><\/math> und offene Intervalle in <math display=\"inline\"><mi>\u211d<\/mi><\/math> sind in folgendem Sinne kompatibel: Ist <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so ist der Schnitt des offenen Balles <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mi>\u211d<\/mi><\/math> gerade das offene, symmetrisch um <math display=\"inline\"><mi>x<\/mi><\/math> liegende Intervall <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>r<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> (wieso?). <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=339&amp;url=Pictures\/Reelle_Zahlen\/komplexe_Ebene\/balls.pdf\"><img decoding=\"async\" id=\"z9229334e039b\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Reelle_Zahlen\/komplexe_Ebene\/balls.svg\" width=\"339\" \/><\/div>  <\/div> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf48355508708\"> <a id=\"x1-61007r53\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.53 <\/span>(Durchschnitt von offenen B\u00e4llen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie folgende Eigenschaft von B<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llen: Seien<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jeden<\/span> <span class=\"ecti-1095\">Punkt <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">existiert<\/span> <span class=\"ecti-1095\">ein Radius <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Illustrieren Sie Ihre Wahl des Radius <\/span><math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">in einem Bild.<\/span> <\/p> <\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zcc9456698ff5\"> <a id=\"x1-61008r54\"><\/a> <span class=\"ecbx-1095\">Definition 2.54 <\/span>(Offene und abgeschlossene Teilmengen von <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Teilmenge <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> heisst <span class=\"ecbx-1095\">offen <\/span>(in <math display=\"inline\"><mi>\u2102<\/mi><\/math>), wenn zu jedem Punkt in <math display=\"inline\"><mi>U<\/mi><\/math> ein offener Ball um diesen Punkt existiert, der in <math display=\"inline\"><mi>U<\/mi><\/math> enthalten ist. Formaler: F\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>U<\/mi><\/math> existiert ein Radius <span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><\/math><\/span><span class=\"period\">.<\/span> Eine Teilmenge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> heisst <span class=\"ecbx-1095\">abgeschlossen <\/span>(in <math display=\"inline\"><mi>\u2102<\/mi><\/math>), falls ihr Komplement <math display=\"inline\"><mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>A<\/mi><\/math> offen ist. <\/p> <\/div> <p class=\"indent\">Nach \u00dcbung <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-61007r53\">2.53<\/a> sind beispielsweise alle B\u00e4lle offen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1c8b1f6e22c7\"> <a id=\"x1-61009r55\"><\/a> <span class=\"ecbx-1095\">Applet 2.55 <\/span>(Offener Ball)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"495px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/t3nn8gv3\/width\/688\/height\/495\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir            sehen,            dass            es            f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r            jeden            Punkt<\/span> <math display=\"inline\"><mi>w<\/mi><\/math> <span class=\"ecti-1095\">in                               dem                               offenen                               Ball<\/span> <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">um<\/span> <math display=\"inline\"><mi>z<\/mi><\/math> <span class=\"ecti-1095\">mit                                                                                                      Radius<\/span> <math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">wieder                                              einen                                              Radius<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gibt,              so              dass              der              offene              Ball              um<\/span> <math display=\"inline\"><mi>w<\/mi><\/math> <span class=\"ecti-1095\">mit                                                                                                      Radius<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">ganz                                                                                                          in<\/span> <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">enthalten ist.<\/span> <\/p> <\/div> <p class=\"indent\">Es gibt, abgesehen von den offenen B\u00e4llen, noch viele weitere, offene Teilmengen von <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Beispielsweise ist jede Vereinigung von offenen Teilmengen offen. Zum Studium der offenen Mengen und damit verwandten Begriffen werden wir in deutlicher gr\u00f6sserer Allgemeinheit im zweiten Semester zur\u00fcckkehren. Insbesondere wollen wir uns hier noch nicht auf eine ausf\u00fchrliche Diskussion einlassen.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-61010r58\"><\/a> <\/p> \n","protected":false},"author":1089,"menu_order":4,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-37","chapter","type-chapter","status-publish","hentry"],"part":33,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/37","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\/37\/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\/37\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=37"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=37"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=37"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=37"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}