From e9db8d4e77060166ac38cd5edfa86669ab173aa4 Mon Sep 17 00:00:00 2001 From: flavis Date: Sun, 1 Dec 2019 12:34:37 +0100 Subject: [PATCH] update ana11, and add ana12 --- ws2019/ana/lectures/analysis11.tex | 11 -- ws2019/ana/lectures/analysis12.pdf | Bin 0 -> 124991 bytes ws2019/ana/lectures/analysis12.tex | 211 +++++++++++++++++++++++++++++ 3 files changed, 211 insertions(+), 11 deletions(-) create mode 100644 ws2019/ana/lectures/analysis12.pdf create mode 100644 ws2019/ana/lectures/analysis12.tex diff --git a/ws2019/ana/lectures/analysis11.tex b/ws2019/ana/lectures/analysis11.tex index 464c4c7..aaba798 100644 --- a/ws2019/ana/lectures/analysis11.tex +++ b/ws2019/ana/lectures/analysis11.tex @@ -147,15 +147,4 @@ Zahlen, weile streng monoton wächst, d.h. $n_1 < n_2 < \ldots.$ bzw. $n_k < n_{ durch Zurückblättern. \end{proof} -\begin{satz}[monoton + beschränkt $\implies$ konvergent] - Eine monoton wachsende (fallende) nach oben (unten) beschränkte Folge konvergiert - gegen ihr Supremum: - \[ - \sup_{n \in \N} a_n := \sup \{a_n | n \in \N\} = \min \{c \in \R | a_n \le c\} - .\] bzw. ihr Infimum: - \[ - \inf_{n \in \N} a_n := \inf \{a_n \mid n \in \N\} = \max \{c \in \R \mid a_n \ge c\} - .\] -\end{satz} - \end{document} diff --git a/ws2019/ana/lectures/analysis12.pdf b/ws2019/ana/lectures/analysis12.pdf new file mode 100644 index 0000000000000000000000000000000000000000..39d5cb305a90b88259dfacf556a0ecab8997d36a GIT binary patch literal 124991 zcmbSyWmKFAwk-}RT!IC+;1m?Dg}b}Egb*A8B)B_4f(G~C7Tn!kf&?dMaEE$yPtV+U zr@8NDUi~O)v5K|6eRQ9F4%A9w5=^Yj902O^>7IE2hzv+(XJQTD=LfLJncG@ATakgd zSUJi5d;wS_-q<*sJCd`z)w!B#=2^cZrFEiml2_LL6sEee3~5C*;)WeQ=ciSPa;hXj@Ri6p zak-j9^}IOUX{%oM_#hF!?}z*2%fnqnho(-2h-0>!aS>(I;QHan!dI>utxoR;B_)b9 zCcyw-u0Wh9B-kHi;dqviLd(6olv1N4hMJjfB#s(5rZj4mvvlezI^ycU%2@8}`^@Rj zw3NIi{+vc6W}{+qkh_N+VPLVigC?$~X{;~r1C`8A%IV}`D}srrFJSL6gLKb}v>})% z9Vh+fW_q00vsIn=!i7u*%;B2Vc;U-^XARjVy`w*x@7KQW$WfDG1liByMVcC_ird+8 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Dann $s - \epsilon$ keine obere + Schranke, d.h. $\exists n_\epsilon \in \N$ mit $s - \epsilon < a_{n_\epsilon}$. + + Damit $s - \epsilon < a_{n_\epsilon} \le a_n \le s < s + \epsilon$ $\forall n \ge n_\epsilon$ \\ + $\implies |a_n - s| < \epsilon$ $\forall n \ge n_\epsilon \implies a_n \to s$ + +\end{proof} + +\begin{satz}[Bolzano-Weierstraß] + Jede beschränkte Folge besitzt mindestens eine konvergente Teilfolge. +\end{satz} + +\begin{proof} + Sei $(a_n)_{n\in\N}$ beschränkt, d.h. $\exists a, b \in \R$, s.d. $a \le a_n \le b$ $\forall n \in \N$. + + Konstruiere induktiv eine Folge von abgeschlossenen Intervallen $I_k := [a_k, b_k]$ mit: + \begin{enumerate}[(1)] + \item $I_k$ enthält unendlich viele Folgenelemente von $(a_n)_{n\in\N}$. + \item $I_k \subset I_{k - 1}$ $\forall k \in \N, k \ge 2$ + \item $(b_k - a_k) \le 2^{1-k} (b_1 - a_1)$ $\forall k \in \N$ + \end{enumerate} + + Für $k = 1$ wähle $a_1 := a$, $b_1 := b$.\\ + $k \to k+1$ : Intervall $I_k := [a_k, b_k]$ mit Eigenschaften (1)-(3) sei konstruiert. + + Berechne $M := \frac{a_k + b_k}{2}$ (Mitte des Intervalls $I_k$). Wegen (1): $[a_k, M]$ + oder $[M, b_k]$ enthält unendlich viele Folgenelemente. + + Setze: + \begin{align*} + I_{k+1} := \begin{cases} + [a_{k}, M] & \text{falls } [a_k, M] \text{ unendlich viele Folgenelemente enthält} \\ + [M, b_k] & \text{falls } [M, b_k] \text{ unendlich viele Folgenelemente enthält} + \end{cases} + \intertext{in beiden Fällen:} + b_{k+1} - a_{k+1} = \frac{b_k - a_k}{2} \stackrel{(3)}{\le } \frac{1}{2} 2^{1-k}(b_1 - a_1) = 2^{-k}(b_1 - a_1) + .\end{align*} $\implies$ (1) - (3) erfüllt für $I_{k+1}$. + + Wir definieren eine Teilfolge $(a_{n_k})$ mit $a_{n_k} \in I_k$ $\forall k \in \N$ :\\ + $k = 1$ : Setze $a_{n_1} := a, n_1 := 1$.\\ + $k \to k+1$ : Wegen (1) ex. ein Index $n_{k+1} > n_k$ mit $a_{n_{k+1}} \in I_{k+1}$. + + $I_k$ bilden eine Intervallschachtelung: + \[ + \implies \underbrace{a_k}_{\to a} \le a_{n_k} \le \underbrace{b_k}_{\to a} \stackrel{\text{Sandwich}}{\implies} a_{n_k} \to a, k \to \infty + .\] +\end{proof} + +\begin{bsp} + $a_n = (-1)^{n}$. Teilfolge: $(1,1,1,1, \ldots) \to 1$, $(-1,-1,-1, \ldots) \to -1$. +\end{bsp} + +\begin{definition}[Häufungspunkt] + Sei $(a_n)_{n\in\N}$ eine Folge in $\R$. Dann heißt $a \in \R$ Häufungspunkt der Folge, + falls $\forall \epsilon > 0$ gilt $|a_n - a| < \epsilon$ für unendlich viele $n \in \N$. +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item $a_n = (-1)^{n}$ hat zwei Häufungspunkte $1$ und $-1$. + \item Falls $\lim_{n \to \infty} a_n = a$, dann ist $a$ Häufungspunkt von $(a_n)_{n\in\N}$. + \end{enumerate} +\end{bsp} + +\begin{bem} + Zu jedem Häufungspunkt $a$ ex. eine Teilfolge $(a_{n_k})_{k \in \N}$ von $(a_n)_{n\in\N}$, die + gegen $a$ konvergiert, also $a = \lim_{k \to \infty} a_{n_k}$: + \[ + a \text{ Häufungspunkt} \iff a = \lim_{k \to \infty} a_{n_k} \text{ für eine } (a_{n_k})_{k \in\N} + .\] +\end{bem} + +\begin{proof} + \begin{enumerate}[(i)] + \item ,,$\implies$'': Sei $a$ Häufungspunkt (HP). Wähle $n_1 \in \N$ mit + $a_{n_1} \in D_1(a) = \{x \mid |x - a| < 1\} $. + + Sei $n_1, \ldots, n_{k-1}$ bereits gewählt.\\ + Wähle $n_k > n_{k-1}$, s.d. gilt: + \[ + a_{n_k} \in D_{\frac{1}{k}}(a) = \left\{x \mid |x-a| < \frac{1}{k}\right\} + .\] + + Dann ist $(a_{n_k})_{k \in\N}$ eine Teilfolge, $|a_{n_k} - a | < \frac{1}{k}$.\\ + $\implies a_{n_k} \to a, k \to \infty$. + \item ,,$\impliedby$ '': Sei $(a_{n_k})_{k\in\N}$ eine Teilfolge mit $\lim_{k \to \infty} a_{n_k} = a$. + + Zu zeigen: $a$ ist HP. + + Sei $\epsilon > 0$. Dann ex. $k_\epsilon \in \N $, s.d. $\forall k \ge k_\epsilon$ gilt: + \[ + |a_{n_k} - a| < \epsilon \implies \forall k \ge k_\epsilon \quad a_{n_k} \in D_\epsilon(a) + .\] + \end{enumerate} +\end{proof} + +\begin{bem} + Satz von Bolzano-Weierstraß besagt, dass jede beschränkte Folge in $\R$ mindestens einen $HP$ besitzt. +\end{bem} + +\begin{definition}[Limes Superior] + Sei $(a_n)_{n\in\N}$ eine Folge in $\R$. Ist $(a_n)_{n\in\N}$ nach oben beschränkt, dann + definiere eine reelle Folge $(s_n)_{n \in N}$ durch $s_n := \text{sup}\{a_k \mid k \ge n\} $. + + $(s_n)_{n\in\N}$ ist monoton fallend. Ist $(s_{n})_{n \in \N}$ nach unten beschränkt, dann definiere + \[ + \lim_{n \to \infty} \text{sup } a_n := \lim_{n \to \infty} s_n = \lim_{n \to \infty} \text{sup}\{a_k \mid k \ge n\} + .\] + + Falls $(a_n)_{n\in\N}$ nicht nach oben beschränkt ist, setzte $\lim_{n \to \infty} \text{sup } a_n := + \infty$. + + Falls $(a_n)_{n\in\N}$ nach oben beschränkt, aber $(s_n)_{n\in\N}$ \textit{nicht} nach unten beschränkt ist, \\ + setze $\lim_{n \to \infty} \text{sup } a_n := - \infty$. +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item $\lim_{n \to \infty} a_n = a \implies \lim_{n \to \infty} \text{sup } a_n = a$. + \item $\lim_{n \to \infty} \text{sup } (-1)^{n} = \lim_{n \to \infty} \text{sup }\{(-1)^{k} \mid k \ge n\} = 1 $ + \item $\lim_{n \to \infty} \text{sup }n = + \infty$\\ + $\lim_{n \to \infty} \text{sup } (-n) = - \infty$ + \end{enumerate} +\end{bsp} + +\begin{definition}[Limes Inferior] + Sei $(a_n)_{n\in\N}$ eine Folge reeller Zahlen. Dann setze: + \[ + \lim_{n \to \infty} \text{inf } a_n := - \lim_{n \to \infty} \text{sup } (-a_n) + .\] +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item $\lim_{n \to \infty} \text{sup } (n^{2}) = + \infty$\\ + $\lim_{n \to \infty} \text{inf } (n^2) = -\lim_{n \to \infty} \text{sup }\{-k^{2} \mid k \ge n\} + = - \lim_{n \to \infty} (-n^2) = +\infty$ + \item \[ + a_n := \begin{cases} + \frac{n}{2} & n \text{ gerade} \\ + 0 & n \text{ ungerade} + \end{cases} + .\] $(a_n) = (0,1,0,2,0,3, \ldots)$ + + $\lim_{n \to \infty} \text{inf } a_n = 0$ \\ + $\lim_{n \to \infty} \text{sup } a_n = + \infty$ + \end{enumerate} +\end{bsp} + +\begin{satz}[Charakterisierung von $\lim \text{sup}$ und $\lim \text{inf}$] + \label{charakterisierung} + Es sei $(a_n)_{n\in\N}$ eine Folge reeller Zahlen. + \begin{enumerate} + \item $\lim_{n \to \infty} \text{sup } a_n = a \in \R \iff$ \\ + $\forall \epsilon > 0$ gilt: + \begin{enumerate}[(i)] + \item $a_n < a+ \epsilon$ für fast alle $n \in N$. + \item $a_n > a - \epsilon$ für unendlich viele $n \in \N$. + \end{enumerate} + \item $\lim_{n \to \infty} \text{inf } a_n = a \in \R \iff$ \\ + $\forall \epsilon > 0$ gilt: + \begin{enumerate}[(i)] + \item $a_n > a - \epsilon$ für fast alle $n \in \N$ + \item $a_n < a + \epsilon$ für unendlich viele $n \in \N$. + \end{enumerate} + \item $(a_n)_{n \in \N}$ ist genau dann konvergent, wenn: + \[ + \lim_{n \to \infty} \text{sup } a_n = \lim_{n \to \infty} \text{inf } a_n = a \in \R + .\] In diesem Fall gilt: $\lim_{n \to \infty} a_n = a$. + \end{enumerate} +\end{satz} + +\begin{bem} + Satz \ref{charakterisierung} impliziert: + \begin{align*} + \lim_{n \to \infty} \text{sup } a_n = \text{sup } \{\text{HP von} (a_n)_{n\in\N}\} \\ + \lim_{n \to \infty} \text{inf } a_n = \text{inf } \{\text{HP von } (a_n)_{n\in\N}\} + .\end{align*} + + $\forall \epsilon > 0$ liegen unendlich viele Folgenelemente im offenen Intervall + \[ + \left(\lim_{n \to \infty} \text{sup } a_n\right) - \epsilon < a_n < \left(\lim_{n \to \infty} \text{sup } a_n\right) + \epsilon \qquad (\text{1 (i) (ii)}) + .\] bzw. + \[ + \left(\lim_{n \to \infty} \text{inf } a_n\right) -\epsilon < a_n < \left( \lim_{n \to \infty} \text{inf } a_n \right) + \epsilon \qquad (\text{2 (i) (ii)}) + .\] + Fast alle Folgenelemente erfüllen: + \[ + \left(\lim_{n \to \infty} \text{inf } a_n\right) - \epsilon < a_n < \left( \lim_{n \to \infty} \text{inf }a_n \right) +\epsilon \qquad (\text{1 (i) und 2 (i)}) + .\] +\end{bem} + +\end{document}