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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?

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    $\begingroup$ If a sequence of real numbers converges then all subsequencee converge to the same limit. $\endgroup$ – Kavi Rama Murthy Feb 23 at 9:01
  • $\begingroup$ @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? $\endgroup$ – Qwaster Feb 23 at 9:03
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    $\begingroup$ 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 $\endgroup$ – Daàvid Feb 23 at 9:42