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?