Montel's theorem

Remarks

Proof idea of part 1: Cauchy's estimates bound the derivatives uniformly on each compact set, so the family is equicontinuous there. The Arzelà–Ascoli theorem on an exhausting sequence of compact sets, with a diagonal argument, gives a convergent subsequence. Part 2, the fundamental normality test, is deeper; one proof uses the modular lambda function, as in the proof of Picard's little theorem. Example: the holomorphic maps $\mathbb{D} \to \mathbb{D}$ of the unit disc form a normal family, since they are bounded by $1$. In part 2 the limit $\infty$ cannot be excluded: the constant functions $f_n = n$ omit $0$ and $1$ for $n \ge 2$ and tend to $\infty$. One omitted value is not enough: the functions $e^{nz}$ on $\mathbb{D}$ omit $0$, yet no subsequence converges near $0$.

Going deeper

For a student

Part 1 is the compactness result behind existence proofs in complex analysis. In the proof of the Riemann mapping theorem, one maximizes $\abs{f'(z_0)}$ over a normal family of injective maps and uses Montel to get a maximizer. Part 2 is the basis of complex dynamics: the Fatou set of a rational map is the set where its iterates form a normal family. It also gives a short proof of Picard's great theorem.