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?
2026-04-23 11:37:28.1776944248
If a sequence converges pointwise and a subsequence converges uniformly does the sequence converge uniformly?
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No: take $(g_n)$ a sequence which converges pointwise but not uniformly say to the null function (where $Y=\mathbb R$) and define $f_{2n}=g_n$, $f_{2n+1}=0$.