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Let $K$ be a compact metric space, and $\{f_n\}_{n \in \mathbb{N}}$ is a uniformly bounded, equicontinuous family of functions. Define $$g_n(x) = \max \{f_1(x),f_2(x),\ldots,f_n(x)\}.$$ Prove that $g_n$ converges uniformly on $K$.

I believe this stems from the fact that $g_n$ is equicontinuous (just take the smallest deltas for $g_n$) and $g_n$ is uniformly bounded. Ergo, it has a convergent subsequence by Arzela-Ascoli. It is then a simple argument to show that if some subsequence of $g_n$ converges, so does the entire sequence.

Is this a fair argument?

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    $\begingroup$ That's look good to me. Probably the hardest part is to show that $g_n$ is equicontinuous. $\endgroup$ – user99914 Sep 24 '15 at 6:24
  • $\begingroup$ You will have to be very careful in choosing your deltas. Though the assumption that $\{f_n:n\in \mathbb N\}$ is equicontinuous should be sufficient to overcome any problems you may have with that. $\endgroup$ – SamM Sep 24 '15 at 7:17
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Hint.

Another proof that doesn't use Arzela-Ascoli theorem.

$(g_n)$ is an increasing sequence of uniformly bounded functions. Hence it converges to a function $g$ that is defined for all points of the compact $K$.

Using equicontinuity, you can prove that $g$ is continuous. You can finally use Dini's theorem or prove it for this specific case.

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