Let $\Omega$ be a bounded subset of $\R^n$ and $(f_k)$ a sequence of functions $f_k\colon \Omega\to \R^m$ 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 $F\subset \mathcal C(X,Y)$ 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 $\mathcal C (X,Y)$ .

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 $\Omega$ 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.