All the proofs of the Ascoli-Arzelà theorem I've encountered so far are really topology dependent, by which I mean the rely deeply on topology concepts such as compactness, open covers, sub covers and so on.
However, I've been told about the existence of a completely analytic proof of this theorem that I have failed to find. Can anyone provide a reference for the proof I am looking for? Thanks in advance.
The version of theorem I'm using is the following:
Let $\{f_n\}_{n \in A}$ be an infinite family of uniformly bounded and equicontinuous functions defined in a bounded set $E \subset \mathbb{R}^d$ and mapping into $\mathbb{R}^m$. Then, there exists a sequence $\{f_k\}_{k\in A}$ that converges uniformly in $E$.