In this document, the following theorem is given:
Theorem 2.1 (Arzela-Ascoli). A subfamily $\mathcal F \subseteq C(U)$ is normal iff it is both equicontinuous and pointwise bounded.
Here, $U\subseteq \mathbb C$ is an open set, and $C(U)$ is the family of continuous functions defined on $U$.
I suspect that the statement of the theorem is not entirely accurate. After all, if $\mathcal F=(f_n)$, where $f_n$ converges uniformly to $\infty$ on $U$, then $\mathcal F$ is a normal family, but it is in no way pointwise bounded.
How could I correct the statement?