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I came across the following problem:

Prove, with out using Arzela-Ascoli, that if $f_n:[0,1]\to\mathbb{R}$ is uniformly equicontinuous and $f_n\to f$ pointwise, then $f_n\to f$ uniformly.

I know how to prove this without using Arzela-Ascoli, but I was wondering how Arzela-Ascoli could be used to prove this in the first place. It only gives that $f_n$ has a subsequence that converges uniformly. Does anyone know?

Tom Chalmer
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    Every subsequence has a further subsequence that converges uniformly to $f$. Hence the full sequence converges uniformly. – Daniel Fischer Jul 24 '17 at 21:07

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