I've come across a question in my Real Analysis course that I'm having quite a lot of difficulty finding an answer to. This post has gotten me the closest to understanding the problem, which is as follows:
"Provide an example or explain why it is not possible: a sequence of functions $(f_n)$ that converges pointwise (but not uniformly) to a function $f$, with a subsequence $(f_{n_k})$ of $(f_n)$ that converges uniformly to $f$."
In general, what conditions need to be satisfied in order for a sequence of functions to have a uniformly convergent subsequence?
Consider : $$ f_{n}(x)=1-\frac{1}{x^{n}+1} $$ Note that $(f_{n}(x))_{n\in\mathbb{N}}$ converges to $f$ point-wise where : $$ f(x)=\begin{cases} 0,& \text{if $x \in[0,1)$} \\ \frac{1}{2}, &\text{if $x=1$} \\ 1,& \text{if $x \in(1,2]$} \end{cases} $$ Each sequence of $f_{n}$ is continuous but the limit function is not continuous. Moreover, we know that if $f_{n}$ is continuous and uniform convergent, then it would imply continuity so if we take the contrapositive statement we can see that no subsequence of $f_{n}$ can converge uniformly to $f$ due to Arzelà–Ascoli theorem