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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]], and let $(u_k)_{k \\ge 1}$ be a sequence of [[subharmonic functions|Z1MHMGFNFW]] on $\\Omega$.\nAssume that the sequence is locally uniformly bounded above: each [[compact set|0R6EWFKRCF]] $L \\subseteq \\Omega$ has a constant $C_L$ with $u_k \\le C_L$ on $L$ for all $k$.\nLet $f$ be a continuous real function on $\\Omega$ with $\\limsup_{k \\to \\infty} u_k(x) \\le f(x)$ for every $x \\in \\Omega$.\nThen for every compact set $K \\subseteq \\Omega$ and every $\\varepsilon \u003e 0$ there is $k_0$ such that\n$$u_k(x) \\le f(x) + \\varepsilon \\qquad \\text{for all } x \\in K \\text{ and all } k \\ge k_0.$$\n\n## Remarks\n\nProof idea: fix $x_0 \\in K$ and a small radius $r$ such that $f \u003c f(x_0) + \\varepsilon/3$ on a closed ball about $x_0$ inside $\\Omega$.\nBy the [[sub-mean value property|BXYQE0Q1ST]], $u_k(y)$ is at most the mean of $u_k$ over the ball $B(y, r)$, for $y$ near $x_0$.\nThe uniform upper bound allows the reverse form of [[Fatou's lemma|S95TR54TCG]], which bounds the [[limit superior|0T3RZQXDYD]] of these ball means by the mean of $\\limsup_k u_k \\le f$.\nThis gives $u_k \\le f(x_0) + \\varepsilon$ on a neighbourhood of $x_0$ for large $k$, and compactness of $K$ finishes the proof.\n\nThe lemma upgrades a pointwise upper bound to a uniform one on compact sets, in the upward direction only.\nA typical use: if $u_k \\le 0$ in the limit at every point, then $\\max_K u_k$ is eventually at most $\\varepsilon$ on each compact $K$.\nThe lower bound is not upgraded in the same way, because subharmonic functions can take very negative values on small sets.\n\n## Explanations\n\n### L1 human\n\nSuppose a sequence of subharmonic functions eventually stays below some continuous ceiling at every single point.\nThen, on any compact piece of the domain, the whole sequence eventually stays below the ceiling plus a little, all at once.\nSubharmonic functions cannot have thin upward spikes, which is why pointwise control becomes uniform control.\n\n### L2 human\n\nThe lemma is named after Hartogs, who used an argument of this kind in his theorem on separately holomorphic functions in several complex variables.\nIt is a basic compactness tool for subharmonic and plurisubharmonic functions.\nTypical uses are bounds on $\\frac{1}{k}\\log\\abs{p_k}$ for sequences of polynomials $p_k$, and the study of upper envelopes of families of subharmonic functions.\n",
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  "name": "Hartogs lemma for subharmonic functions",
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