Let \(\Omega\subseteq\CC\) be open and let \(\mcf\) be a family of holomorphic functions on \(\Omega\) that is uniformly bounded on compact subsets of \(\Omega\). Then
\(\mcf\) is equicontinuous on every compact subset of \(\Omega\), and
\(\mcf\) is a normal family.
Let \(K\subseteq\Omega\) be compact and choose \(r>0\) with \(3r<\operatorname{dist}(K,\CC\sm\Omega)\) (any \(r>0\) if \(\Omega=\CC\)). The set \(K'\coloneqq\ts{z\st\operatorname{dist}(z,K)\le2r}\) is a compact subset of \(\Omega\), so there is \(M\) with \(\abs f\le M\) on \(K'\) for all \(f\in\mcf\). For \(z,w\in K\) with \(\abs{z-w}<r\), the circle \(C=\ts{\abs{\zeta-w}=2r}\) lies in \(K'\), and the Cauchy integral formula gives \[ \abs{f(z)-f(w)}=\abs{\frac{1}{2\pi i}\int_Cf(\zeta)\Big(\frac{1}{\zeta-z}-\frac{1}{\zeta-w}\Big)d\zeta}\le\frac{1}{2\pi}\cdot2\pi(2r)\cdot M\cdot\frac{\abs{z-w}}{r\cdot2r}=\frac{M}{r}\abs{z-w}, \] which is independent of \(f\); this is equicontinuity on \(K\). Normality follows from the Arzelà–Ascoli theorem applied on an exhaustion of \(\Omega\) by compact sets, with a diagonal argument.
Montel’s second theorem uses the extended convention for normality in the remark on D-QTJ7T, where a subsequence may instead tend to \(\infty\) uniformly on compact subsets: a family of meromorphic functions on a region \(\Omega\) that omits the same three values of the Riemann sphere is normal in that sense.