Arzela-Ascoli_Theorem

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Definition

If $\{f_n\}$ is a uniformly bounded and equicontinuous sequence of functions on a compact set $K$, then there exists a subsequence $\{f_{n_k}\}$ that converges uniformly on $K$.

A family of functions is equicontinuous if for every $\epsilon > 0$ there exists $\delta > 0$ such that $|f_n(x) - f_n(y)| < \epsilon$ for all $n$ whenever $|x - y| < \delta$.