1
$\begingroup$

Assume that $K$ is a compact metric space, real-valued functions $g_n$ defined on $K$ form an equicontinuous family and $g_n(x)$ is convergent for each $x\in K$. It follows by the Ascoli-Arzela theorem that some sequence $g_{k_n}$ of $g_n$ is convergent uniformly.

How to obtain using Ascolli-Arzela theorem that the sequence $(g_n)$ is convergent uniformly?

$\endgroup$
1
$\begingroup$

Your second assumption ($g_n(x)$ converges for every $x$) implies that each (uniformly) convergent subsequence necessarily converges to the pointwise limit $g(x) := \lim_n g_n(x)$.

If $g_n$ would not converge to $g$ the there would be a subsequence which does not, and to that subsequence you can apply Arzela Ascoli again to arrive at a contradiction.

| cite | improve this answer | |
$\endgroup$

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.