# Question about pointwise convergence vs. uniform convergence

1. Mar 13, 2009

### 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?

2. Mar 13, 2009

### 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.

3. Mar 14, 2009

### HallsofIvy

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