One of the ways the Arzela-Ascoli Theorem is stated is as follows:
Given a compact space $X$ and a set $ M \subset C(X) := \{f: X \rightarrow \mathbb{R}, \| . \|_{\infty}\}$, the following are equivalent:
1) $M$ is bounded, closed and uniformly equicontinuous.
2) $M$ compact.
Why is the condition on boundedness required, and does it not follow from uniform equicontinuity? It appears to me that uniform equicontinuity implies (pointwise) continuity of any function $f$ in $M$, and since $X$ is compact and $f$ is continuous, $f(X)$ is compact so $f$ takes its maximum and minimum on its image. In particular, it would follow that $\| f\|_{\infty} < \infty$, for every $f \in M$, so $M$ is bounded. Since the theorem is well-established, it seems to me that there must be some mistake in this argument.