# proof of Ascoli-Arzelà theorem

Given $\epsilon>0$ we aim at finding a $4\epsilon$-net in $F$ i.e. a finite set of points $F_{\epsilon}$ such that

 $\bigcup_{f\in F_{\epsilon}}B_{4\epsilon}(f)\supset F$

(see the definition of totally bounded). Let $\delta>0$ be given with respect to $\epsilon$ in the definition of equi-continuity (see uniformly equicontinuous) of $F$. Let $X_{\delta}$ be a $\delta$-lattice in $X$ and $Y_{\epsilon}$ be a $\epsilon$-lattice in $Y$. Let now $Y_{\epsilon}^{X_{\delta}}$ be the set of functions from $X_{\delta}$ to $Y_{\epsilon}$ and define $G_{\epsilon}\subset Y_{\epsilon}^{X_{\delta}}$ by

 $G_{\epsilon}=\{g\in Y_{\epsilon}^{X_{\delta}}\colon\exists f\in F\ \forall x% \in X_{\delta}\quad d(f(x),g(x))<\epsilon\}.$

Since $Y_{\epsilon}^{X_{\delta}}$ is a finite set, $G_{\epsilon}$ is finite too: say $G_{\epsilon}=\{g_{1},\ldots,g_{N}\}$. Then define $F_{\epsilon}\subset F$, $F_{\epsilon}=\{f_{1},\ldots,f_{N}\}$ where $f_{k}\colon X\to Y$ is a function in $F$ such that $d(f_{k}(x),g_{k}(x))<\epsilon$ for all $x\in X_{\delta}$ (the existence of such a function is guaranteed by the definition of $G_{\epsilon}$).

We now will prove that $F_{\epsilon}$ is a $4\epsilon$-lattice in $F$. Given $f\in F$ choose $g\in{Y_{\epsilon}}^{X_{\delta}}$ such that for all $x\in X_{\delta}$ it holds $d(f(x),g(x))<\epsilon$ (this is possible as for all $x\in X_{\delta}$ there exists $y\in Y_{\epsilon}$ with $d(f(x),y)<\epsilon$). We conclude that $g\in G_{\epsilon}$ and hence $g=g_{k}$ for some $k\in\{1,\ldots,N\}$. Notice also that for all $x\in X_{\delta}$ we have $d(f(x),f_{k}(x))\leq d(f(x),g_{k}(x))+d(g_{k}(x),f_{k}(x))<2\epsilon$.

Given any $x\in X$ we know that there exists $x_{\delta}\in X_{\delta}$ such that $d(x,x_{\delta})<\delta$. So, by equicontinuity of $F$,

 $d(f(x),f_{k}(x))\leq d(f(x),f(x_{\delta}))+d(f_{k}(x),f_{k}(x_{\delta}))+d(f(x% _{\delta}),f_{k}(x_{\delta}))<4\epsilon.$
Title proof of Ascoli-Arzelà theorem ProofOfAscoliArzelaTheorem 2013-03-22 13:16:19 2013-03-22 13:16:19 paolini (1187) paolini (1187) 12 paolini (1187) Proof msc 46E15