Theorem (Arzelà-Ascoli)

Let $(X,d)$ be a Compact Metric Space and $F \subset C(X)$. Then F is Totally Bounded for $d_{sup}$ if and only if it is Uniformly Bounded and Equicontinuous, thus uniformly equicontinuous, i.e. $\forall\epsilon>0$ $$\begin{gather*} |F|<\infty:\forall f \in C(X),\exists g\in F : \sup_{x \in \mathbb{X}}|f(x)-g(x)|<\epsilon\\ \iff\\ \exists\delta>0:d(x,y)<\delta\implies|f(x)-f(y)|<\epsilon,\forall f\in \mathscr{F} \end{gather*}$$