Question about pointwise convergence vs. uniform convergence

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

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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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