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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be an open set and let $\\mathcal{F}$ be a family of [[harmonic functions|0DEB0JMN0W]] on $U$.\nAssume that $\\mathcal{F}$ is uniformly bounded on each compact subset of $U$.\nThat is, for each compact $K \\subseteq U$ there is $M_K$ with $\\abs{u(x)} \\le M_K$ for all $u \\in \\mathcal{F}$ and $x \\in K$.\nThen every sequence in $\\mathcal{F}$ has a subsequence that converges uniformly on each compact subset of $U$, and the limit is harmonic on $U$.\n\n## Remarks\n\nProof idea: the [[gradient estimates|N76R7JWMNJ]] bound $\\abs{\\nabla u}$ on each compact set, uniformly for $u \\in \\mathcal{F}$, so the family is [[equicontinuous|0MW2WXTHMC]] on compact sets.\nThe [[Arzelà–Ascoli theorem|0381S8Z5P0]] on an exhaustion of $U$ by compact sets, with a diagonal argument, gives a subsequence converging uniformly on compact sets.\nThe limit is continuous and inherits the mean value property, so it is harmonic by the [[converse of the mean value property|CQ7E0V6T2G]].\nThe derivatives of all orders then converge uniformly on compact sets as well.\nFor nonnegative harmonic functions on a connected $U$, it is enough that the family is bounded at a single point, by [[Harnack's inequality|YNH832D50W]]; this is closely related to [[Harnack's principle|P15K8RKHDV]].\nThe boundedness hypothesis is needed: the constants $u_k = k$ have no convergent subsequence.\nExample: the family $\\set{e^{t x} \\cos(t y) : \\abs{t} \\le 1}$ on $\\R^2$ is uniformly bounded on compact sets, so every sequence in it has a locally uniformly convergent subsequence.\n\n## Explanations\n\n### L1 human\n\nA collection of harmonic functions that stays within fixed bounds on each bounded region cannot oscillate wildly.\nFrom any sequence in it, one can extract a subsequence that settles down to a limit, and that limit is again harmonic.\n\n### L2 human\n\nThis is the harmonic analogue of Montel's theorem on normal families of holomorphic functions.\nIt is the compactness tool behind existence proofs in potential theory, such as the construction of Green's functions by exhaustion and of extremal harmonic functions.\nOn a connected domain, Harnack's inequality upgrades boundedness at one point to local boundedness for positive families.\nSo the positive harmonic functions with $u(x_0) = 1$ form a compact set, and this compactness underlies the Martin boundary.\n",
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