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zaTPIPRS{JQ31JB_ArWzT!2fP?zm68v)y~r%ASNREFJcY&%e$RD?*jn>s?PV(24$!X z+!p$mwj(_6V-hYzK8uV~X6oQqRCwzRpQB-kFzBM^Z zBS*}XLy{w-LJ9P7-gDKk$FV1DvZwQLy#HD#{ktz9Me)hw=%|L}7%2^G_YW$zpQ3HrhP5O5xB;$?lJRtPP9b?SowKx8)F7zA)A5PA2I0tMhEX(oE^!J=OB@DT6A z6;1(3@F)Ui!E2%#q7kVnJvR@+Ym~=cvJ6250ygNsW#P))2A8kB-o~#kdvf6QR*u{j z=OB=*7VxLAfJOF8mk2lD(Nla3`qwo&?NbHOVJd6d$4`>0?N_6 zj%DE7Bu#JrD=EWM&+Orzap@ac(E#k!*6vMmMAj!;ToVn$1i{*j&@xd|!4~c;t+%j7 zuTA`BX)@E#T@7Hj?Ci!`5BdxD4QhKb)^>b}pb)e1E)GJi+HVJVcT3xh5oL;-IQPQpjD!H@yHu-ZjhMga>29NktR+TFxpou%Frdy zWr}mQ$THCye9fg*OWCvsh|^HR|D}+n;!E}a9)E 0$ mit $b + \delta = a$. + + Wegen der Konvergenz: + \[ + b_n \to b, a_n \to a, n \to \infty + .\] + \[ + \exists n_\epsilon \in \N \text{ sd. } |b - b_n| \le \frac{1}{2} \delta + .\] und + \[ + |a - a_n| \le \frac{1}{2} \delta \text{ } \forall n > n_\epsilon + .\] + Dann + \[ + b_n = b_n - b + b - a + a - a_n + a_n \le |b_n - b| + b - a | a - a_n| + a_n + \le \frac{1}{2} \delta - \delta + \frac{1}{2} \delta + a_n = a_n + .\] $\implies$ + \[ + b_n \le a_n + .\] Widerspruch zur Annahme, dass $a_n \le b_n$ für fast alle $n \in \N$. +\end{proof} + +\begin{bem}[Folgerung aus 3] + Sei Cauchy-Folge $(a_{n})_{n \in \N} $ keine Nullfolge und + $a_n \to a$, $a > 0$ $n \to \infty$. + Dann $a_n > 0$ für fast alle $n$. +\end{bem} + +\begin{proof} + Annahme: $a_n \le 0$ für fast alle $n \in N$, dann + $a_n \to a \le 0$ $\leftarrow$ $\{0\}$ $n \in N$ +\end{proof} + +\textbf{Ziel}: Reelle Zahlen als Grenzwerte von rationalen Cauchy Folgen. + +Wichtig: Zwei Cauchy Folgen mit gleichem Limes definieren gleiche Zahl. +Deshalb: \underline{Äquivalenzklassen} + +\begin{definition}[Äquivalenzrelation für Cauchy Folgen rationaler Zahlen] + \begin{align*} + (a_n)_{n\in\N} \thicksim (a'_n)_{n\in\N} :\iff |a_n - a'_n| \to 0, n \to \infty + .\end{align*} +\end{definition} + +\begin{proof}[Die Relation ist Äquivalenzrelation] + + \begin{enumerate} + \item Reflexivität $(a \sim a)$ (trivial) + \item Symmetrie $(a \sim b \implies b \sim a)$ + \[ + (a_n)_{n\in\N} \sim (b_n)_{n\in\N} \iff |a_n - b_n| \to 0 + \iff |b_n - a_n| \to 0 + \iff (b_n)_{n\in\N} \sim (a_n)_{n\in\N} + .\] + \item Transitivität $a \sim b, b \sim c \implies a \sim c$ + \begin{align*} + (a_n)_{n\in\N} \sim (b_n)_{n\in\N}, (b_n)_{n\in\N} ~ (c_n)_{n\in\N} + \iff |a_n - b_n| \to 0, |b_n - c_n| \to 0 \\ + \iff \forall \epsilon > 0 \exists n_\epsilon \text{ s.d. } \\ + \forall n \ge n_\epsilon + .\end{align*} + Dann + \[ + |a_n - c_n| = |(a_n - b_n) + (b_n - c_n)| \\ + \le |a_n - b_n| + .\] + \end{enumerate} +\end{proof} + +\begin{definition}[Äquivalenzklassen] + \begin{align*} + \overline{\R} :=& \{ [a_n]_{n \in \N}\} \\ + =& \{ (a'_n)_{n\in\N} | (a'_n)_{n\in\N} \sim (a_n)_{n\in\N}\} \\ + =& \{ (a'_n)_{n\in\N} | (a_n' - a_n)_{n \in \N} \to 0\} + .\end{align*} + $(a_n)_{n\in\N}$ Repräsentant von Klasse $[(a_n)]_{n \in \N}$ +\end{definition} + +\begin{bem}[] + $a \in \Q \implies$ + \[ + [(a_n)_{n\in\N}, a_n := a ] \in \overline{\R} + .\] + Jede Teilfolge $(a_{n_k})_{{k}\in\N}$ einer Cauchy Folge + \[ + (a_{n_k})_{{k}\in\N} \in [(a_n)_{n\in\N}] + .\] + Jede Äquivalenzklasse $[\left( (a_n)_{n\in\N} \right) ]$ von Cauchy + Folge rationaler Zahlen definiert genau eine reelle Zahl +\end{bem} + +\begin{satz}[] + Jeder Äquivalenzklasse $[(a_n)_{n\in\N}]$ entspricht genau einem + (möglicherweise unendlichen) Dezimalbruch. + + Die Menge aller dieser Dezimalbrüche wird bezeichnet als Menge + $\R$ der ,,reellen Zahlen''. + \[ + \R = \left\{a := \pm (a_0 + 0,d_1d_2d_3\ldots d_k\ \mid + a_0 \in \N_0, d_k \in \{0, \ldots, 9\} \right\} + .\] + Für eine CF rationaler Zahlen $(a_n)_{n\in\N}$ wird $a \in \R$ + als Grenzwert bezeichnet: + \[ + a = \lim_{n \to \infty} a_n + .\] + $(a_n)_{n\in\N}$ heißt eine ,,approximierende'' Folge von $a \in \R$. + In diesem Sinne hat jede CF rationaler Zahlen nach Konstruktion + einen Grenzwert in $\R$. +\end{satz} + +\begin{bem}[Erinnerung Geometrische Reihe] + \[ + 1 + x + x^2 + \ldots + x^{n} = \frac{1-x^{n+1}}{1 - x}, x + 1 + .\] +\end{bem} + +\begin{proof} + \begin{enumerate} + \item $\forall a \in \R$ $\exists [(a_n)_{n\in\N}] \in \overline{\R}$ + \[ + a = \pm (a_0 + 0,d_1d_2d_3 \ldots ) + .\] + definieren wir eine Folge $(a_n)_{n\in\N}$ (rationaler) endlicher + Teilbrüche: + \[ + a_n = \pm (a_0, d_1\ldots d_n), a_0 \in \N_0, d_k \in \{0, \ldots, 9\} + .\] zu zeigen: $(a_n)_{n\in\N}$ ist eine Cauchy Folge. + + Sei $m > n + 1$, dann + \begin{align*} + |a_n - a_m| &= |a_0 + 0,d_1d_2\ldots d_n - + (a_0 + 0, d_1d_2 \ldots d_n d_{n+1} \ldots d_m )| \\ + &= | 0,00\ldots 0 d_{n + 1} \ldots d_m| \\ + &= d_{n+1} 10^{-(n+1)} + \ldots + d_m 10^{-m} \\ + &\le 10^{-n} (d_{n+1} 10^{-1} + \ldots + d_m 10^{-m+n}) \\ + &\le 10^{-n} (10^{0} + \ldots + 10^{-m+n+1}) \\ + &= 10^{-n} \left( \left( \frac{1}{10}^{0} \right) + + \ldots + \frac{1}{10}^{m-n-1} \right) \\ + &= 10^{-n} \frac{1 - \frac{1}{10}^{m-n}}{1 - \frac{1}{10}} \\ + &\le 10^{-n} \frac{1}{\frac{9}{10}} \\ + &= 10^{-n} \frac{10}{9} \to 0, n \to \infty + .\end{align*} + $\implies (a_n)_{n\in\N}$ ist Cauchy Folge und repräsentiert + eine Klasse $[(a_n)_{n\in\N}] \in \overline{\R}$ + + ,,Einbettung'' $a \mapsto [(a_n)_{n\in\N}]$ + + \item Wir zeigen, dass diese ,,Einbettung'' bijektiv ist. + + \textbf{a)} $a \mapsto [(a_n)_{n\in\N}]$ ist injektiv + ($\forall a, a' \in \overline{R}$ gilt: + aus $(a_n)_{n\in\N} \sim (a'_n)_{n\in\N}$ folgt $a = a'$) + + Für + \begin{align*} + &a = a_0 + 0,d_1d_2 \ldots \\ + &a' = a'_{0} + 0,d'_1d'_2 \ldots + \end{align*} + gilt: + \begin{align*} + |a_n - a'_n| &= |a_0 + 0,d_1\ldots d_n - (a'_0 + 0, d'_1 \ldots d'_n)| \\ + &\le \epsilon, \forall n \ge n_\epsilon, \forall \epsilon > 0 + .\end{align*} + $\implies a = a'$ + + \textbf{b)} ,,Einbettung'' $a \mapsto [(a_n)_{n\in\N}]$ ist + surjektiv + + \begin{enumerate}[(i)] + \item Sei $(a_n)_{n\in\N}$ Nullfolge + \[ + \implies z = 0 = 0,00 \ldots + .\] + \item Sei $(a_n)_{n\in\N}$ keine Nullfolge + Dann fast alle $a_n > 0$ oder fast alle + $a_n < 0 $ + O.B.d.A. $a_n > 0$, $n \in \N$ + + Ziel: $z \ge 0$ zu konstruieren. + + Falls $a_n < 0$ (bzw. $-a_n > 0$ konstruiert $-z$) + \[ + (a_n)_{n\in\N} \implies \text{beschränkt} \\ + \implies \exists N \in \N (N \ge 2) \text{ s.d. } + 0 < a_n < N, n \in \N + .\] + + Dann $\exists z_0 \in \N_0$, s.d. im Interval: + \[ + I_0 := \{x \in \Q \mid 0 \le z_0 \le x < z_0 + 1 < n\} + .\] unendlich viele Elemente von $(a_n)_{n\in\N}$ liegen. + \end{enumerate} + \end{enumerate} +\end{proof} +\end{document}