Equicontinuity and uniformly convergent subsequences

Question

I have been trying to grasp equicontinuity lately and with it, the Arzela-Ascoli Theorem. It says that if ${f_k}$ is a sequence of functions on a compact interval that is uniformly bounded and uniformly equicontinuous, then there exists a uniformly convergent subsequence.

I recently read a post (at this link: Sequence $(f_n)$ equibounded, equicontinuous but with not uniformly convergent subsequence), where it was shown that some sequence of functions was uniformly bounded and equicontinuous, but there was no uniformly convergent subsequence.

I am thoroughly confused, since this seems to contradict the AA-theorem. I just keep seeing more examples of uniformly bounded and equicontinuous sequences of continuous functions that do not have any uniformly convergent subsequences. If someone could clear up the confusion, it would be fantastic.

Thanks!

Answer 1

To understand why compactness of the domain is necessary, consider the following. Let $f: \mathbb{R} \to \mathbb{R}$ be any continuous, bounded function supported on the interval $[0,1]$. Since $[0,1]$ is compact, $f$ is uniformly continuous. Let $f_{k}(x) = f(x - k)$, i.e. translating $f$ to the right by $k$. Then all of the functions $f_{k}$ are bounded, and they obviously uniformly equicontinuous, because they all share the same uniform continuity properties. However, this sequence of functions cannot have a uniformly convergent subsequence, because every subsequence converges pointwise to the zero function, but does not uniformly converging to the zero function. Indeed, if $f(x) \neq 0$ for $x \in [0,1]$, then $f_{k}(x+k) = f(x)$ is not getting arbitrarily close to $0$, no matter how large $k$ is.