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?