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?

  • 1
    $\begingroup$ 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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.