Question about pointwise convergence vs. uniform convergence

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The discussion confirms that if a sequence of functions \{f_n\} converges pointwise to 0 on the entire real line, any subsequence \{f_{n_k}\} that converges uniformly must also converge to 0. This is due to the fact that uniform convergence entails pointwise convergence, ensuring all subsequences converge to the same limit. Therefore, the limiting function f of the subsequence must indeed be 0.

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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?
 
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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.
 
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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