Question about pointwise convergence vs. uniform convergence

AxiomOfChoice

Suppose you know that a sequence $\{f_n\}$ of functions converges pointwise to 0 on the whole real line. If there is a subsequence $\{f_{n_k}\}$ of the original sequence that converges uniformly to a limiting function $f$ on the whole real line, does that limiting function have to be 0?

owlpride

Yes. Uniform convergence implies pointwise convergence, and when a sequence of points converges to a limit point, all subsequences will converge to that same limit.

HallsofIvy

Homework Helper
In fact, if a sequence coverges pointwise, then every subsequence converges to the same thing, whether that convergence is pointwise or uniform.

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