Under which hypothesis is a pointwise convergent sequence of functions, with a uniformly convergent subsequence, uniformly convergent?

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.

• Take $f_1=f_3=f_5=\cdots=0$ identically, and $f_2,f_4,f_6,\ldots$ being a sequence tending to zero pointwise but not uniformly. – Lord Shark the Unknown Mar 10 '18 at 13:53
• If any subsequence has a uniformly convergent subsequence, then already the sequence have to be uniformly convergent. – p4sch Mar 10 '18 at 13:57