TITLE: Uniform convergence of subsequences implying uniform convergence
QUESTION [7 upvotes]: Suppose ${f_n}$ are uniformly bounded and equicontinuous on some closed interval $[a,b]$. Therefore, by Arzela-Ascoli we know that $f_n$ has a uniformly convergent subsequence. 
But we can also apply Arzela-Ascoli to the subsequences of $f_n$ to get that every subsequence of $f_n$ has a subsequence that converges uniformly. And we have some extra information that says each of the subsequences of subsequences converges uniformly to the same thing.
How do you conclude that $f_n$ converges uniformly from this?

REPLY [9 votes]: Let $f$ be the common limit of the uniformly convergent subsequence. Suppose $f_n$ does not converge uniformly to $f$, so we have some $\epsilon>0$ and a collection of points $x_{n_k}\in [a,b]$ such that $n_k\to \infty$ and $\|f_{n_k}(x_{n_k})-f(x_{n_k})\|>\epsilon$. But then no subsequence of $f_{n_k}$ can converge uniformly to $f$, a contradiction. Hence $f_n$ converges uniformly to $f$.