# 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

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

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving