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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be an open set and let $\\mathcal{F}$ be a family of [[holomorphic|0DG0Z72CQD]] functions $U \\to \\C$.\nThe family $\\mathcal{F}$ is *normal* if every sequence $(f_n)$ in $\\mathcal{F}$ has a [[subsequence|0R8J3ZQMJS]] that converges [[uniformly|08HJRTG5N7]] on each [[compact|0QPHW703PV]] subset of $U$.\n\n## Remarks\n\nExample: $\\mathcal{F} = \\set{z^n : n \\ge 1}$ on the unit disc $\\mathbb{D}$ is normal.\nOn a compact set $K \\subset \\mathbb{D}$ there is $r \u003c 1$ with $\\abs{z} \\le r$ on $K$, so $\\abs{z^n} \\le r^n \\to 0$ uniformly on $K$.\nNon-example: $\\set{n z : n \\ge 1}$ on $\\mathbb{D}$ is not normal.\nEach subsequence tends to $\\infty$ at each $z \\neq 0$ but stays $0$ at $z = 0$, so it does not converge on a disc around $0$.\nThe limit of a convergent subsequence is holomorphic, by Weierstrass's theorem that [[uniform limits of holomorphic functions are holomorphic|0431RN0R1N]].\nMany authors also accept a subsequence that tends to $\\infty$ uniformly on compact subsets.\nWith this convention, and with the chordal metric of the [[Riemann sphere|0CDMXGPSES]], the definition extends to families of meromorphic functions.\n",
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