Doubt on Arzela-Ascoli theorem

Question

Consider a sequence of equicontinuous and uniformly bounded functions on a compact set. Under which condition I can say that it has a unique uniformly convergent subsequence ? Or, atleast uniform convergent subsequences with the same unique limit ?

Answer 1

Answer. A sequence $\{f_n\}_{n\in\mathbb N}\subset\mathscr F$, where $\mathscr F$ a uniformly bounded and equicountinuous family on a compact set, possesses a unique uniformly convergent subsequence iff the sequence $\{f_n\}_{n\in\mathbb N}$ is itself uniformly convergent.

Explanation. For if not, $\{f_n\}_{n\in\mathbb N}$ would would still have a convergent subsequence $f_{k_n}\to f$, but as $f_n\not\to f$, there would be an $\varepsilon>0$, such that for every $n_0>0$, there exists a $n_1>n_0$, with $\|f_{n_1}-f\|\ge\varepsilon$. This allows us to construct a subsequence $\{f_{\ell_n}\}$, with $\|f_{\ell_n}-f\|\ge\varepsilon$, for all $n$. But Arzela-Ascoli provides a uniformly convergent subsequence of $\{f_{\ell_n}\}$, say converging to $g$, and clearly $\|f-g\|\ge\varepsilon$, and thus $f\ne g$. So $\{f_n\}_{n\in\mathbb N}$ has to be uniformly convergent.

Answer 2

The sequence of functions will have a unique limit if and only if it converges uniformly. Otherwise it will have multiple distinct accumulation points.