Ascoli-Arzelà theorem

Let $\mathrm{\Omega }$ be a bounded subset of ${ℝ}^n$ and $(f_k)$ a sequence of functions $f_k:\mathrm{\Omega }\to {ℝ}^m$. If $\{f_k\}$ is equibounded and uniformly equicontinuous then there exists a uniformly convergent subsequence $(f_{k_j})$.

A more abstract (and more general) version is the following.

Let $X$ and $Y$ be totally bounded metric spaces and let $F\subset 𝒞(X,Y)$ be an uniformly equicontinuous family of continuous mappings from $X$ to $Y$.Then $F$ is totally bounded (with respect to the uniform convergence metric induced by $𝒞(X,Y)$).

Notice that the first version is a consequence of the second.Recall, in fact, that a subset of a complete metric space is totally bounded if and only if its closure is compact (or sequentially compact).Hence $\mathrm{\Omega }$ is totally bounded and all the functions $f_k$ have image in a totally bounded set. Being $F=\{f_k\}$ totally bounded means that $\overline{F}$ is sequentially compact and hence $(f_k)$ has a convergent subsequence.