Is a uniformly bounded sequence of functions pointwise bounded?

Question

I thought that if some sequence of functions is uniformly bounded, then it should be pointwise bounded.

But I saw the following:

If {$f_n$} is pointwise bounded on $E$ and $E_1$ is a countable subset of $E$, it is always possible to find a subsequence {$f_{n_k}$} such that {$f_{n_k}(x)$} converges for every $x \in E_1$. However, even if {$f_n$} is a uniformly bounded sequence of continuous functions on a compact set E, there need not exist a subsequence that converges pointwise on E.

If I was correct, a uniformly bounded sequence of continuous functions should be a pointwise bounded sequence, then I should always be possible to find a subsequence converging, but it's not true.

I wonder where I am wrong, and also want to know some counterexamples that is pointwise bounded functional sequence but not uniformly. Thank you in advance.

Yes, they are pointwise bounded. So for each point $x$ you can find a subsequence such that $f_{n_k}(x)$ is Cauchy. If you work hard with the diagonal argument, given a countable set $S$ such can find a subsequence $f_{n_k}$ such that, for all $x\in S$, $f_{n_k}(x)$ is Cauchy. However, $E$ may be uncountable, so those arguments don't work and in fact you may not find a subsequence that converges on the whole $E$. — Sassatelli Giulio, Oct 22 '22 at 17:14
Oh, I got it. The statement is saying that there need not exist a subsequence converging on whole $E$, even if it is uniformly bounded, but of course I can find converging subsequence on countable subset $E_1$ when it is uniformly bounded also, am I right? — SeungJae Bang, Oct 22 '22 at 17:19
Does this answer your question? Does uniform boundedness imply pointwise boundedness? Note also that if you have a counterexample $E$, with a sequence of functions $f_n:E\to[-1,1]$ having no pointwise convergent subsequence, this gives you an example of space $[-1,1]^E$ which is compact (by Tychonov) but not sequentially compact. — Anne Bauval, Oct 22 '22 at 17:23
For such a counterexample, see Henno Brandsma's answer in Product space that is compact, but isn't sequentially compact. — Anne Bauval, Oct 22 '22 at 17:40

Is a uniformly bounded sequence of functions pointwise bounded?

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