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zDFxdhPRT(0jQ~6$fR%%q5N!<& N7@vZIsf`)mzW@|e*ogoD literal 0 HcmV?d00001 diff --git a/ws2019/ana/lectures/analysis8.tex b/ws2019/ana/lectures/analysis8.tex new file mode 100644 index 0000000..4b6de9e --- /dev/null +++ b/ws2019/ana/lectures/analysis8.tex @@ -0,0 +1,228 @@ +\documentclass{../../../lecture} + +\begin{document} + +\begin{bem}[Organisatorisches] + Nächsten Mittwoch (20. November) findet keine Vorlesung und auch keine Plenarübung statt. Aber Mittwoch (27.11.) Vorlesung + statt Plenarübung im großen Hörsaal Chemie +\end{bem} + +\begin{satz}[Wichtiger Satz] + Der Körper $\R$ ist vollständig, d.h. jede + Cauchy Folge in $\R$ hat einen Limes +\end{satz} + +\begin{proof} + Sei $(a_n)_{n\in\N}$ eine Cauchy Folge reeller Zahlen, d.h. $a_n \in \R$. + \[ + a_n \in \R: \forall n \in \N: \exists \text{ C.F. } (a_{n, m}) + .\] + \[ + a_{n,m} \in \Q \forall n, m \in \N, a_n = \lim_{m \to \infty} a_{n,m} + .\] + + $\forall n \in \N$ wähle Index $k_n \in \N$ mit + \[ + |a_n - a_{n,kn}| < \frac{1}{n} + .\] + + $k_n$ existiert, weil $\lim_{m \to \infty} a_{n,m} = a_n$ und damit $|a_n - a_{n,m}| \to 0$ also $\exists \epsilon |a_n - a_{n.m}| < \epsilon < \frac{1}{n}$ + und Archimedisches Axiom. + + Ziel: zu zeigen $\left( a_{n, k_{n}} \right)_{n \in \N}$ rationaler Zahlen ist Cauchy Folge. + + Sei $\epsilon > 0$. Dann $\exists n_{\epsilon} \in \N$ s.d. $\forall n, m \ge n_\epsilon$ : + \begin{align*} + |a_n - a_m| < \frac{1}{3} \epsilon, |a_n - a_{n,k_n}| < \frac{1}{3} \epsilon \\ + (a_n)_{n \in \N} \text{C.F.} \\ + |a_m - a_{m,k_m}| < \frac{1}{3} \epsilon \text{(AA)} + .\end{align*} + + und folglich + \[ + |a_{n, k_n} - a_{m, k_m}| \le |a_{n, k_n} - a_n + a_n - a_m + a_m - a_{m, k_m}| \le |a_{n, k_n} - a_n| + | a_n - a_m| + |a_m - a_{m, k_m}| < \epsilon + .\] + $\implies (a_{n, k_n})_{n \in \N}$ Cauchy Folge + $\implies$ Nach Konstruktion der $\R$ folgt, dass $\exists $ ,,limes' $a \in \R$, s.d. + \[ + \forall \epsilon > 0, \exists n_\epsilon \in \N, \forall n \ge n_\epsilon: |a_{n, k_n} - a| < \epsilon + .\] + Dann gilt für die Folge $(a_n)_{n\in\N}$: + \[ + |a_n - a| \le |a_n - a_{n, k_n}| + |a_{n, k_n} - a| \le \frac{1}{n} + |a_{n, k_n} - a| \to 0 + .\] + $\implies a = \lim_{n \to \infty} a_n$ + + $\Q$ ist dicht in $\R$, d.h. + \[ + \forall a \in \R \text{ gilt } \forall \epsilon > 0 \exists q_\epsilon + \in \Q \text{ s.d. } |a - q_\epsilon| \le \epsilon + .\] + + Nach Konstruktion von $\R$ folgt: + \begin{align*} + \forall \text{C.F.} (a_n)_{n\in\N} a_n \in \Q: \\ + \exists a \in \R: a = \lim_{n \to \infty} a_n + .\end{align*} + $\implies \forall \epsilon > 0$ $\exists n_\epsilon$ $|a_n - a| < \epsilon \forall n \ge n_\epsilon$ +\end{proof} + +\begin{bem}[Archimedisches Axiom] + \begin{align*} + \forall a \in \R: \exists n \in \N: \text{s.d.} n - a > 0 \\ + \implies \forall \epsilon > 0 \exists n \in \N \text{s.d. } n - \frac{1}{\epsilon} > 0 \\ + \implies \frac{1}{n} < \epsilon + .\end{align*} +\end{bem} + +\subsection{Wichtige Aussage} + +$\R$ ist vollständig, da alle C.F. in $\R$ haben einen Grenzwert in $\R$. + +\subsection{Weitere Möglichkeiten, die Vollständigkeit von $\R$ zu charakterisieren} + +\begin{definition}[Maximum, Minimum, obere/untere Schranke, Supremum, Infimum] + Sei $M \subset \R, M \neq \emptyset$. + + Maximum: + \[ + \text{max} M := b \in M: b \ge x, \forall x \in M + .\] + Minimum: + \[ + \text{min} M := a \in M: x \ge a, \forall x \in M + .\] + Obere Schranke: + \[ + b \in \R, \text{ s.d. } b \ge x, \forall x \in M + .\] + Untere Schranke: + \[ + a \in \R, \text{ s.d. } x \ge a, \forall x \in M + .\] + Eine Menge $M$ heißt nach oben (unten) beschränkt, falls eine obere (untere) Schranke von $M$ existiert. + + Supremum: kleinste obere Schranke + + Infimum: größte untere Schranke +\end{definition} + +\begin{bsp} + \begin{itemize} + \item $\N$ ist von unten beschränkt, z.B. mit 0. $\text{min} \N = 1$, von oben unbeschränkt. + \item $M = \{x \in \R \mid x^2 < 2\} $: obere Schranke ist $\sqrt{2} $ und untere Schranke ist + $ - \sqrt{2} $, aber $M$ besitzt kein Maximum bzw. Minimum. + $\sqrt{2}$ ist Supremum und $-\sqrt{2} $ ist Infimum + \end{itemize} +\end{bsp} + +\begin{bem} + Falls $b = \text{sup} M \iff$: + \begin{enumerate} + \item $b$ ist eine obere Schranke von $M$, d.h. $\forall x \in M: x \le b$ und + \item Jede Zahl $c < b$ ist keine obere Schranke von M, d.h. $\forall c \in M, c < b, \exists x \in M$: $c < x$ + (oder $\forall \epsilon > 0 \exists x \in M\colon x > b - \epsilon$) + \end{enumerate} + + Analog für $a = \text{inf}M$ +\end{bem} + +\begin{bsp} + \[ + I = (a, b) := \{x \in \R \mid a < x < b\} + .\] dann gilt $\text{sup} I = b, \text{inf} I = a$. +\end{bsp} + +\begin{bem} + Das Supremum (Infimum) muss nicht zur Menge $M$ gehören, aber falls + $\text{sup}M \in M$, dann $\text{sup}M = \text{max} M$. +\end{bem} + +\begin{satz}[Vollständigkeit in $\R$ Nr. 2] + $\R$ vollständig $\iff$ jede nichtleere beschränkte Teilmenge $M \in \R$ besitzt ein Supremum bzw. Infimum +\end{satz} + +\begin{definition}[Intervalle] + \[ + [a, b] := \{x \in \R \mid a \le x \le b\} \text{ abgeschlossenes Intervall} + .\] + \[ + (a, b) := \{x \in \R \mid a < x < b\} \text{ offenes Intervall} + .\] + \[ + (a, b] := \{x \in \R \mid a < x \le b\} \text{ halboffenes Interval} + .\] + \[ + [a, b) \text{ analog} + .\] +\end{definition} + +\begin{definition}[Intervallschachtelung] + ist eine Folge von abgeschlossenen Intervallen $I_n := [a_n, b_n] := \{ x \in \R \mid a_n \le x \le b_n\}, n \in \N $ + mit Eigenschaften. + 1) $I_{n+1} \subset I_n n \in \N$ (bedeutet $a_n \le a_{n+1} \le b_{n+1} \le b_n$ + 2) $\forall \epsilon > 0, \exists I_{n}$ mit der Länge + \[ + |b_n - a_n| < \epsilon \text{ d.h. } |b_n - a_n| \to 0, n \to \infty + .\] +\end{definition} + +\begin{satz}[Vollständigkeit in $\R$ Nr. 3] + Vollständigkeit in $\R$ $\iff$ Intervallschachtelungseigenschaft d.h. für jede Intervallschachtelung. + \[ + (I_n)_{n\in\N} \in \R, \exists c \in \R + \] so dass + \[ + {c} = \bigcap_{n = 1}^{\infty} I_n := \{x \in \R | x \in I_n \forall n \in \N\} + .\] + Diese Aussage ist verwandt mit dem Axiom vom Dedekindischen Schnitt +\end{satz} + +\begin{satz}[Trennungseigenschaft] + Seien $A, B \subset \R, A \neq \emptyset, B \neq \emptyset$ mit + $a < b \forall a \in A, b \in B$ + + Dann existiert immer ein $c \in \R$, welches A und B trennt: + \[ + \forall a \in A, b \in B \text{ gilt } a \le c \le b + .\] + Dies ist ebenfalls $\iff$ zur Vollständigkeit in $\R$ +\end{satz} + +\begin{lemma}[Existenz der $k$-ten Wurzel einer positiven reellen Zahl] + $\forall a \in \R^{+}$ $\forall k \in \N$: existiert eine positive $k$-te Wurzel. + Das heißt die Lösung der Gleichung + \[ + x^{k} = a + .\] ist $\sqrt[k]{a}$ (Bezeichnung). +\end{lemma} + +\begin{proof} + 1) Die Eindeutigkeit der $\sqrt[k]{a}$ (falls sie existiert) + + Seien $x_1, x_2 \in \R$ zwei $k$-te Wurzeln des $a \in R^{+}$ : + \[ + x_1^{k} = a = x_2^{k} + .\] + Dann gilt: + \[ + 0 = x_1^{k} - x_2^{k} = (x_1 - x_2) \underbrace{\sum_{m=0}^{k-1} x_1^{k-1-m} x_2^{m}}_{> 0} + .\] mit dieser Hilfsformel + \[ + x^{n} - y^{n} = (x - y) \sum_{k=0}^{n-1} x^{n-1-k}y^{k} + .\] $\implies$ $x_1 - x_2 = 0 \implies x_1 = x_2$ + + 2) Existenz: + $a = 1 \implies \sqrt[k]{1} = 1$ ($1^{k} = 1$ ) + Sei $a > 1$ und Annahme, dass $\exists$ Wurzel für $0 < a' < 1$ + Dann definiere: + \[ + \sqrt[k]{a} := \frac{1}{\sqrt[k]{\frac{1}{a}} } + .\] + \[ + \left( \sqrt[k]{a} \right) ^{k} = \left( \frac{1}{\sqrt[k]{a'} } \right)^{k} = \frac{1}{\sqrt[k]{a'}^{k} } = \frac{1}{a'} = a + .\] + Es bleibt zu zeigen: $\exists \sqrt[k]{a} $ für $0 < a < 1$. +\end{proof} + +\end{document}