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  "body": "\n## Statement\n\nLet $U \\subseteq \\C$ be an open set and let $\\mathcal{F}$ be a family of [[holomorphic|0DG0Z72CQD]] functions $U \\to \\C$.\n\n1. Assume that $\\mathcal{F}$ is locally uniformly bounded: for each compact $K \\subset U$ there is $M_K$ with $\\abs{f(z)} \\le M_K$ for all $f \\in \\mathcal{F}$ and $z \\in K$. Then $\\mathcal{F}$ is a [[normal family|QWSD2JDRP0]].\n2. Assume that $U$ is a [[domain|097KNJFJ7V]] and that there are two distinct complex numbers $a$ and $b$ that no $f \\in \\mathcal{F}$ takes as a value. Then each sequence in $\\mathcal{F}$ has a subsequence that converges uniformly on compact subsets of $U$, either to a holomorphic function or to $\\infty$.\n\n## Remarks\n\nProof idea of part 1: [[Cauchy's estimates|0JJRENWMP6]] bound the derivatives uniformly on each compact set, so the family is equicontinuous there.\nThe [[Arzelà–Ascoli theorem|0381S8Z5P0]] on an exhausting sequence of compact sets, with a diagonal argument, gives a convergent subsequence.\nPart 2, the fundamental normality test, is deeper; one proof uses the [[modular lambda function|4Z1YXT957Z]], as in the proof of [[Picard's little theorem|R2PHM118NG]].\nExample: the holomorphic maps $\\mathbb{D} \\to \\mathbb{D}$ of the unit disc form a normal family, since they are bounded by $1$.\nIn 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 $\\infty$.\nOne omitted value is not enough: the functions $e^{nz}$ on $\\mathbb{D}$ omit $0$, yet no subsequence converges near $0$.\n\n## Explanations\n\n### L1 human\n\nMontel's theorem says that a family of holomorphic functions cannot be wild if it is bounded.\nFrom any sequence you can always extract a subsequence that converges nicely.\nFor real functions this is false: $\\sin(nx)$ is bounded, but no subsequence converges uniformly.\nThe second part says that even missing two values is enough to tame the family.\n\n### L2 human\n\nPart 1 is the compactness result behind existence proofs in complex analysis.\nIn the proof of the [[Riemann mapping theorem|02WHMEG5ZR]], one maximizes $\\abs{f'(z_0)}$ over a normal family of injective maps and uses Montel to get a maximizer.\nPart 2 is the basis of complex dynamics: the Fatou set of a rational map is the set where its iterates form a normal family.\nIt also gives a short proof of [[Picard's great theorem|08QA8QTM71]].\n",
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