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 $E$.

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    $\begingroup$ Have you looked at Rudin's Principles of Mathematical Analysis? $\endgroup$ – Ted Shifrin Oct 8 '16 at 16:25
  • $\begingroup$ What version of the Ascoli-Arzelà are you talking about? The statement itself in the general version uses the topological concepts. I would not expect that the proof can be completely analytic in such version. $\endgroup$ – Jack Oct 8 '16 at 16:29
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    $\begingroup$ @Jack I've edited the question with the version of the theorem I'm using. $\endgroup$ – user313212 Oct 8 '16 at 16:36
  • $\begingroup$ @TedShifrin I haven't, but I'll take a look and see what I can find. Thanks! $\endgroup$ – user313212 Oct 8 '16 at 16:36
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    $\begingroup$ It doesn't exist in that book by Rudin, but it does in "Real and Complex Analysis". Yet the language is "tainted" with topological concepts and, besides this, it is given for complex functions. The proof though is not that messy and without much topology. $\endgroup$ – DonAntonio Oct 8 '16 at 16:59

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