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Allora sia $f : E \to \QQ$ tale che $\xbar \mapsto c$, dove $c \in \QQ$ è + un punto razionale in $(L^-(\xbar), L^+(\xbar))$ (tale $c$ esiste sempre, per la densità di $\QQ$ in $\RR$). + Inoltre, $x < y \implies L^+(x) \leq L^-(y)$, e quindi ogni intervallo da cui è preso $c$ è distinto al variare + di $\xbar \in E$. Quindi $f$ è iniettiva, e vale $\abs{E} \leq \abs{\QQ} = \abs{\NN}$. Si conclude allora + che $E$ è al più numerabile. + \end{solution} \begin{theorem} (della permanenza del segno) Data $(x_n) \subseteq \RR$ tale che $x_n \tendston L > 0$, allora @@ -414,7 +435,21 @@ la tesi. \end{proof} - %TODO: aggiungere dimostrazione alternativa con il metodo della bisezione. + \begin{proof}[Dimostrazione alternativa] (metodo di bisezione per la ricerca degli zeri) + Come prima, senza alcuna perdita di generalità, si pone $f(a) < 0 < f(b)$. Si ponga $x_0 = \frac{a+b}{2}$, + $I_0 = (a, b)$. + Se $f(x_0) = 0$, allora il teorema è dimostrato. Altrimenti, $f(x_0) > 0$ o $f(x_0) < 0$. Nel primo caso, + si consideri $I_1 = (a, x_0)$, altrimenti si ponga $I_1 = (x_0, b)$. Si riapplichi allora l'algoritmo con $a := \inf I_1$ e $b := \sup I_1$, definendo la successione $(x_n)$ e gli intervalli $I_n$ per ogni passo $n$ dell'algoritmo. \\ + + Se la successione $(x_n)$ è finita, allora $\exists n \mid f(x_n) = 0$, e quindi il teorema è dimostrato. Altrimenti, + si osservi che la successione degli intervalli è decrescente, e che $\abs{I_n} = \frac{b-a}{2^n} \tendston 0$: + allora, poiché $x_n \in I_n$ $\forall n \in \NN$, $(x_n)$ ammette limite. In particolare, $I_n \tendston \{c\}$, + e quindi $x_n \tendston c \in I_0$. Siano $a_n$, $b_n$ le successioni tali che $I_n = (a_n, b_n)$ $\forall n \in \NN$. + Vale in particolare che $a_n, b_n \tendston c$. Allora, per la continuità di $f$ su $(a, b)$, vale che + $\lim_{n \to \infty} f(a_n) = f(c)$ e che $\lim_{n \to \infty} f(b_n) = f(c)$: poiché ogni elemento di $(a_n)$ è + per costruzione tale che $f(a_n) < 0$, deve valere che $f(x) \leq 0$ per il teorema della permanenza del segno; analogamente deve valere per costruzione di $(b_n)$ che $f(c) \geq 0$. Si conclude allora che $f(c) = 0$, da cui + la tesi. + \end{proof} \begin{corollary} (dei valori intermedi) Dati $I = (a, b)$ e $f : I \to \RRbar$ continua, allora $y_1$, $y_2 \in f(I) \implies