# If a sequence converges pointwise and a subsequence converges uniformly does the sequence converge uniformly?

Let $f_{n}: X \rightarrow Y$ be a sequence of continuous functions from one metric space to another. Suppose the sequence converges pointwise. However a subsequence $\{ f_{n_k} \}$ converges uniformly. Can one conclude that that the sequence converges uniformly?

No: take $(g_n)$ a sequence which converges pointwise but not uniformly say to the null function (where $Y=\mathbb R$) and define $f_{2n}=g_n$, $f_{2n+1}=0$.