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  "body": "\n## Statement\n\nLet $A_1, A_2, \\dots$ be [[events|00SQGJNRTB]] in a probability space $(\\Omega, \\mathcal{F}, P)$ with $\\sum_{n=1}^{\\infty} P(A_n) \u003c \\infty$.\nThen\n$$P\\left(\\limsup_{n \\to \\infty} A_n\\right) = 0, \\quad \\text{where} \\quad \\limsup_{n \\to \\infty} A_n = \\bigcap_{n=1}^{\\infty} \\bigcup_{m=n}^{\\infty} A_m.$$\nSo almost surely, only finitely many of the events $A_n$ occur.\n\n## Remarks\n\nThe event $\\limsup_n A_n$ is the set of outcomes that lie in infinitely many $A_n$. It is written $\\{A_n \\text{ i.o.}\\}$, for \"infinitely often\".\nProof: for each $n$, $P(\\limsup_m A_m) \\le P\\left(\\bigcup_{m \\ge n} A_m\\right) \\le \\sum_{m \\ge n} P(A_m)$ by [[Boole's inequality|0EGSEE9J9R]]. The right side tends to $0$ as $n \\to \\infty$.\nThe lemma needs no independence. The [[second Borel–Cantelli lemma|0NWXRAMEE7]] gives a partial converse for independent events.\nThe lemma holds in each measure space: if $\\sum_n \\mu(A_n) \u003c \\infty$, then $\\mu(\\limsup_n A_n) = 0$.\n\n## History\n\nThe lemma is named after Émile Borel and Francesco Paolo Cantelli.\n\n## Explanations\n\n### L1 human\n\nIf the chances of the events add up to a finite number, then after some time none of them happens any more, with probability $1$.\nFor example, if $P(A_n) = 1/n^2$, then only finitely many $A_n$ occur.\n",
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