uniform convergence of subsequences

Question

I have the following question. Given a sequence of functions $f_n(x)$ for which there is a subsequence of functions uniformly converging to some $\overline{f}(x)$. What can we say about $f_n(x)$? What additional assumption is needed to ensure uniform convergence of $f_n(x)$? Thanks!

Answer 1

There is a subsequence converging uniformly to $\overline{f}(x)$, but this says nothing at all about the members of the sequence that are not in the subsequence.

If you want the whole sequence to converge, you need some assumption that gives you control over the other members of the sequence. For example, it would be sufficient that every subsequence has a subsequence that converges uniformly to $\overline{f}(x)$.

Answer 2

Not much can be said about $f_n$; for example, consider $f_n(x)=n$ if $n$ is odd and $f_n(x)=1/n$ if $n$ is even. The subsequence $f_{2n}$ clearly converges uniformly to $0$, but $f_n$ doesn't even converge pointwise.