Let $C(X)$ be the space of continuous functions with the usual norm. $X$ be a compact metric space. The Arzela Ascolis theorem says:
A subset $S$ of $C(X)$ is compact iff it is uniformly bounded and equicontinuous at any point of $x$.
I think the uniform boundedness can be relaxed to that $S$ is bounded at any point $x$.
Is that actually true or are there prominent counter examples?