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  "body": "\n## Statement\n\nLet $G$ be a [[domain|097KNJFJ7V]] in $\\mathbb{C}$, that is, a connected open set.\nLet $(f_k)$ be a sequence of [[holomorphic|0DG0Z72CQD]] functions on $G$ that converges uniformly on each compact subset of $G$ to a function $f$.\nSuppose that $f$ is not identically zero on $G$, and that $f$ has a zero of [[order|064N002TX4]] $m$ at a point $z_0 \\in G$.\nThen for each sufficiently small $\\rho \u003e 0$ there is an integer $N$ with the following property.\nFor each $k \\ge N$, the function $f_k$ has exactly $m$ zeros in the disc $|z - z_0| \u003c \\rho$, each counted as many times as its order.\n\n## Remarks\n\nThe limit $f$ is holomorphic on $G$ by the [[theorem on uniform limits|0431RN0R1N]].\nA consequence: if no $f_k$ has a zero in $G$, then $f$ either has no zero in $G$ or is identically zero.\nA second consequence: if each $f_k$ is injective on $G$, then $f$ is injective or constant.\nThe radius $\\rho$ must be small: $f_n(z) = z - 1 + \\frac{1}{n}$ has one zero in the unit disc, but its limit $z - 1$ has none there.\n\n## Explanations\n\n### L1 human\n\nZeros of holomorphic functions do not suddenly appear or disappear under uniform limits.\nNear a zero of the limit, each function far along in the sequence has the same number of zeros.\nThe proof compares $f_k$ with $f$ on a small circle, as in [[Rouché's theorem|0D430QR02W]].\n",
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