[Math] Arzela-Ascoli Theorem: Is only pointwise boundedness required

Question

In Royden's text the Arzela-Ascoli Theorem states:

Let X be a compact metric space and $f_n$ a uniformly bounded, equicontinuous sequence of real valued functions on X. Then $f_n$ has a subsequence that converges uniformly on X, to a continuous function f on X.

However I cannot seem to see where the hypothesis of uniform boundedness is used in the proof – it seems that only pointwise boundedness of the sequence is required. My question: is uniform boundedness actually required, or could we replace "uniformly bounded" with "pointwise bounded", in the statement of the theorem? And if we cannot: what is an example of a pointwise but not uniformly bounded sequence for which the theorem fails?

Answer 1

(Migrated from the comment) Recall that uniform continuity on a totally bounded set (for instance, a compact set) implies that a function is bounded.

Similarly, for a family $\mathcal{F}$ of functions on a metric space $X$,

1. Equicontinuity of $\mathcal{F}$,
2. Totally-boundedness of $X$, and
3. Pointwise boundedness of $\mathcal{F}$

implies that the family of function is uniformly bounded. So if you assume that $X$ is compact and $(f_n)$ is equicontinuous, then $(f_n)$ is pointwise bounded iff it is uniformly bounded.