# Does every convergent subsequence of a pointwise convergent function sequence converge to the same limit? [closed]

Let $$\left(f_n\right)_{n \in \mathbb{N}}$$ be a function sequence that 1.) converges pointwise 2.) uniformly to limiting function $$f$$. Is it true that every convergent subsequence of $$f_n$$ must converge to the same limit, whether $$f_n$$ converges uniformly or only pointwise?

• If a sequence of real numbers converges then all subsequencee converge to the same limit. – Kavi Rama Murthy Feb 23 at 9:01
• @KaviRamaMurthy How about with function sequences and with pointwise/uniform convergence? If the "main" sequence converges pointwise, does this imply that any subsequence will also converge pointwise to the same limit? – Qwaster Feb 23 at 9:03
• You said that it converges pointwise. Then to each point in the domain of the $f_{n}$'s, you will have a subsequence, $f_{\alpha (n)}(x)$ which converges to $f(x)$ since it is a sequence of real numbers. But then the subsequence converges pointwise to $f$, and that's it. So pointwise convergence in the original sequence estabilishes pointwise convergence for subsequences, but the same also holds trivially for uniform convergence – Daàvid Feb 23 at 9:42