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]

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