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+\begin{document} + +\textbf{Heute:} Frustcafé (deswegen kürzere Plenarübung) + +\textbf{Nächsten Mittwoch:} Vorlesung fällt aus, aber Ersatztermin +wird gesucht. + +\section{Reelle Zahlen} + +Fortsetzung Beweis: + +\begin{proof} + \begin{enumerate} + \item Zz: $\forall [ (a_n)_{n\in\N}] \in \overline{\R}$ $\exists z \in \R$ + \[ + z = \pm \left( z_0 + 0,d_1d_2d_3 \ldots ) \right) + .\] + O.B.d.A. $z > 0, a_n > 0, n \in \N$ + \[ + (a_n)_{n\in\N} \text{ C.F. } \implies 0 < a_n < N + \text{ } \forall n \in \N + .\] $\implies z_0 \in \N_0$, s.d. + O.B.d.A. $(a_n)_{n\in\N} \subset I_0$ + \[ + I_0 := \{ x \in \Q \mid 0 \le z_0 \le x < z_0 + 1 < N\} + .\] + \begin{tikzpicture} + \draw (0, 0) -- (10, 0); + \end{tikzpicture} + + $I_0$ wird unterteilt in 10 Teilintervalle. + + Für ein $d_1 \in \{0, \ldots, 9\} $ ein Intervall + \[ + I_1 := \{ x \in I_0 \mid z_0 + d_1 \cdot 10^{-1} + \le x < z_0 + (d_1 + 1) \cdot 10^{-1}\} + .\] + Sei $z_1 = z_0 + 0,d_1 $, dann + \[ + I_1 = \{ x \in I_0 \mid z_1 \le x < z_1 + 10^{-1}\} + .\] + $\implies \exists n_1$ Index s.d. $|z_1-a_{n_1}| \le 10^{-1}$ + + usw. $\ldots$ + + Ergebnis: eine Folge von Teilintervallen + + o.B.d.A. + \[ + (a_n)_{n\in\N} \subset \ldots \subset I_{k+1} \subset I_k \subset \ldots \subset I_1 \subset I_0 + .\] + \[ + z_k := z_{k-1} + d_k \cdot 10^{-k} \in \Q + .\] + \[ + I_k = \{x \in I_{k-1} \mid z_k \le x < z_k + 10^{-k}\} + .\] + $\exists n_k:$ Index s.d. $|z_k - a_{n_k}| \le 10^{-k}$ + + Das heißt für eine Folge + \[ + z_k := z_0 + 0,d_1d_2 \ldots d_k \in \Q, k \in \N + .\] + existiert eine Teilfolge $(a_{n_k})_{k\in\N}$ von der + C.F. $(a_n)_{n\in\N}$, s.d. $|z_k - a_{n_k}| \le 10^{-k}$, $k \in \N$ + $\implies (z_k - a_{n_k})_{k \in \N}$ Nullfolge, d.h. \\ + $\implies (z_k)_{k\in\N} \sim (a_{n_k})_{k\in\N}$ \\ + $\implies (z_k)_{k\in\N} \sim (a_n)_{n\in\N} $ \\ + $\implies (z_k)_{k\in\N} \in [(a_n)_{n\in\N}]$ + + und der resultierende Dezimalbruch ist: + \[ + z := z_0 + 0,d_1d_2 \ldots d_k \ldots \in \R + .\] + \end{enumerate} + + Wir haben gezeigt: + \[ + \forall [(a_n)_{n\in\N}] \in \overline{\R} \text{: } \exists z \in \R + .\] Damit: $\implies$ Abbildung ist surjektiv und damit bijektiv. + \[ + \implies \exists \text{ ,,inverse Abbildung''}: \overline{\R} \to \R + .\] die auch bijektiv ist. + + Diese Abbildung ist auch verträglich mit der Addition und der + Multiplikation, d.h. $[(a_n)_{n\in\N}] \mapsto a$ und $[(a_n')_{n \in \N} \mapsto a']$ + + Dann $[ (a_n)_{n\in\N}] + [(a'_n)_{n\in\N}] := a + a'$ \\ + $[ (a_n)_{n\in\N}] \cdot [(a'_n)_{n\in\N}] := a \cdot a'$ + + Abbildung $\R \longleftrightarrow \overline{\R}$ ist Isomorphismus. +\end{proof} + +\begin{bem} + + Die Darstellung ist durch einen Dezimalbruch ist nicht immer eindeutig. + z.B.: + \[ + 0,9999 \ldots = 0,\overline{9} = 1 = 1,0000 \ldots + .\] + Deshalb, falls $z = z_0 + 0,d_1d_2 \ldots d_k 999 $ $d_k \le 8$ dann: + + $z := z_0 + 0, d_1 d_2 \ldots (d_k + 1) 0 \ldots$ +\end{bem} + +\begin{bem} + Der Satz gilt auch für ,,b-adische'' Brüche mit Basis $b \in \N, b \ge 2$ : + $a \in \R$ besitzt eine sogenannte ,,b-adische Entwicklung'': + \[ + a = \pm (a_0 + 0,d_1d_2 \ldots) = \pm (a_0 + d_1 \cdot b^{-1} + d_2 \cdot b^{-2} + \ldots ) + .\] mit $a_0 = g_0 + g_1 \cdot b + g_2 \cdot b^{2} + \ldots$ $\in \N_0$ + mit Ziffern $d_n, g_n \in \{0, 1, \ldots, b-1\} $ + Für $b=2$ : dijadische Entwicklung +\end{bem} + +\subsubsection{Zusammenfassung} + +Beobachtung: Jede reelle Zahl ist ein Grenzwert von einer Folge +$(a_n)_{n\in\N}$ rationaler Zahlen. + +Beispiel: $\sqrt{2} $ + +\begin{tikzpicture} + \draw (0, 0) -- (10, 0) -- (10, 0.5) -- (0, 0.5) -- (0,0); + \draw (2, 0) -- (8, 0) -- (8, 0.3) -- (2, 0.3) -- (2,0); + \draw (4, 0) -- (6, 0) -- (6, 0.2) -- (4, 0.2) -- (4,0); +\end{tikzpicture} + +$\forall (a_n), (b_n)$ $a_n \to a, b_n \to a \iff (a_n - b_n)_{n \in \N}$ Nullfolge. + +Deshalb: + +\begin{itemize} + \item Definiere C.F. rationaler Zahlen + \item Äquivalenzrelation: + \[ + (a_n)_{n\in\N} \sim (b_n)_{n\in\N} :\iff (a_n - b_n)_{n \in \N} \text{ Nullfolge } + .\] + und Äquivalenzklasse: + \[ + \overline{R} := \{ [(a_n)_{n\in\N}]\} + .\] + \item Eine Klasse aus $\overline{\R} \iff$ eine reelle Zahl +\end{itemize} + +Konstruktion nach Cantor, 1873 + +Als nächstes: $\R$ ist ein angeordneter Körper mit ,,$+$ '', ,,$\cdot$'', +,,$>$'' und ist auch ,,vollständig''. + +\subsection{Der Körper $\R$} + +Seien $a \in \R, b \in \R$ und $(a_n)_{n\in\N}$, $(b_n)_{n\in\N}$ +zugehörige approximierende Folgen rationaler Zahlen. + +Alle Struktureigenschaften von $\Q$ sind über den Grenzübergang +auf $\R$ übertragbar. + +\begin{definition}[Absolutbetrag] + \[ + |a| := \lim_{n \to \infty} |a_n| + .\] + Folglich: Begriffe ,,Konvergenz'' und ,,Cauchy-Folgen'' gelten auch + für Folgen reeller Zahlen. +\end{definition} + +\begin{definition}[Arithmetische Grundoperationen] + \[ + a + b := \lim_{n \to \infty} (a_n + b_n) + .\] + \[ + a \cdot b := \lim_{n \to \infty} \left( a_n \cdot b_n \right) + .\] +\end{definition} + +\begin{definition}[Ordnungsrelation] + \[ + a > b :\iff \lim_{n \to \infty} (a_n - b_n) > 0 + .\] + und folglich: $\exists \alpha \in \Q_+$ s.d. $a_n - b_n \ge \alpha$ für + fast alle $n \in \N$. + \[ + a \ge b :\iff a > b \text{ oder } a = b + .\] + \[ + a < b :\iff b > a + .\] + \[ + a \le b :\iff b \ge a + .\] +\end{definition} + +\begin{definition}[Positivität] + \[ + \R^{+} := \{a \in \R \mid a > 0\} + .\] +\end{definition} + +\begin{bem} + Definitionen sind unabhängig von der Wahl der Folge: + + \begin{proof}[Beispiel für Absolutbetrag] + Seien $(a_n)_{n\in\N}, (a'_n)_{n\in\N}$ zwei approximierende + Folgen von $a$, d.h. $(a_n - a'_n)_{n \in \N}$ ist eine Nullfolge. + + Zu zeigen: + \[ + |a| = \lim_{n \to \infty} |a_n| = \lim_{n \to \infty} |a'_n| + .\] d.h. zu zeigen: + \begin{align*} + &\lim_{n \to \infty} |a_n| = \lim_{n \to \infty} |a'_n| \\ + \iff &\lim_{n \to \infty} \left( |a_n| - |a'_n|\right) = 0 + .\end{align*} + + Betrachte: + \[ + | |a_n| - |a'_n| | \le | a_n - a_n'| < \epsilon + .\] $\implies \left( |a_n| - |a'_n| \right) $ ist Nullfolge \\ + $\implies |a_n| = |a_n'|$ \\ + $\implies \lim_{n \to \infty} |a_n| = \lim_{n \to \infty} |a'_n| = |a|$ + \end{proof} + + Die anderen Beweise folgen analog. +\end{bem} + +\begin{satz}[Der vollständige Körper $\R$] + \begin{enumerate} + \item $(\R, +, \cdot, >)$ ist angeordneter Körper + \item $\Q$ ist Unterkörper von $\R$ + \item Der Körper $\Q$ ist vollständig, d.h. jede Cauchy-Folge in + $\R$ hat einen Grenzwert in $\R$. + \item Der Unterkörper $\Q$ ist ,,dicht'' in $\R$, d.h. + \[ + \forall a \in \R, \forall \epsilon > 0, \exists g_\epsilon \in \Q \text{ s.d. } |a - q_\epsilon| < \epsilon + .\] + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[1)] + \item Körperaxiome (Kommutativität, Assoziativität, Distributivität) + sind trivial + + Neutrales Element der Addition: + \[ + 0 := 0, 0\ldots \text{ Klasse der Nullfolgen z.B.: } + a_n = 0 \forall n \in \N + .\] + Neutrales Element der Multiplikation + \[ + 1 := 1,0 \ldots 0 \text{\hspace{10mm}} [(1)_{n \in\N}] + .\] + Existenz der inversen Elemente bezüglich der Addition + \[ + a + x = 0, a \in \R + .\] + \[ + x = -a = \lim_{n \to \infty} (- a_n) + .\] + + Existenz der inversen Elemente bezüglich der Multiplikation + \[ + b \cdot x = 1, b \in \R \setminus \{0\} + .\] + \[ + x = \frac{1}{b} := \lim_{n \to \infty} c_n + .\] + \[ + (c_n)_{n\in\N} = \text{ ? } + .\] + $b \in \R \setminus \{0\} \implies \lim_{n \to \infty} b_n = b \neq 0$ \\ + $ \implies (b_n)_{n\in\N}$ keine Nullfolge $\implies$ fast + alle Elemente $b_n \neq 0$ (Übung!) + + + Definiere: + \[ + c_n := \begin{cases} + 0 & b_n = 0 \\ + \frac{1}{b_n} & b_n \neq 0 + \end{cases} + .\] + Dann $(b_n \cdot c_n) $ = + \[ + (b_n \cdot c_n) = \begin{cases} + 0 & b_n = 0 \\ + 1 & b_n \neq 0 + \end{cases} + .\] + $\implies \lim_{n \to \infty} (b_n \cdot c_n) = 1$ + \item $a \in \Q$ entspricht $[(a_n)_{n \in \N}]$ mit + $a_n = a$ $\forall n \in \N$ $\implies$ $\Q \subset \R$, + $\Q$ Körper \\ + $\implies Q$ Unterkörper von $\R$. + \item Sei $(a_n)_{n\in\N}$ eine C.F. reeller Zahlen. + \[ + \forall a_n \in \R \text{ } \exists \text{ approx. Folge} (a_{n, m})_{m \in \N} + .\] + \[ + a_{n, m} \in \Q \forall n, m \in \Q + .\] + \[ + a_n := \lim_{n \to \infty} a_{n, m} n \in \N + .\] + $\forall n \in \N$ wähle $k_n \in \N$ mit: + \[ + |a_n - a_{n, k_{n}}| < \frac{1}{n} + .\] Wir zeigen, dass $(a_{n, k_n})_{n \in \N}$ rationaler Zahlen eine + C.F. ist. + + Sei $\epsilon > 0$. Dann + \end{enumerate} +\end{proof} + +\end{document}