I was trying to undestand the relation between pointwise and uniform convergence of sequences and subsequences of functions and the following question popped up:
What conditions must be assumed over a sequence of functions $f_n$ to guarantee that if it converges pointwise to a function $f$ and if it has a subsequence $f_{n_k}$ that converges uniformly, then $f_n$ is itself uniformly convergent?
I'm also looking for some counterexample showing when this does not hold, I think it might help me to understand.