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  "body": "\n## Proof\n\nWe bound the limit superior by the tails of the convergent series.\n\n1. Let $L = \\limsup_{n} A_n = \\bigcap_{n \\ge 1} B_n$, where $B_n = \\bigcup_{m \\ge n} A_m$. Each $B_n$ and $L$ are [[events|00SQGJNRTB]], as countable unions and intersections of events.\n2. For each $n$, $L \\subseteq B_n$. So $P(L) \\le P(B_n)$, by the monotonicity of the measure $P$ on the [[probability space|0ZYBZ34RT2]].\n3. By [[Boole's inequality|0EGSEE9J9R]], $P(B_n) \\le \\sum_{m = n}^{\\infty} P(A_m)$.\n4. Let $s = \\sum_{m=1}^{\\infty} P(A_m)$, a finite number by hypothesis. The tail $\\sum_{m \\ge n} P(A_m) = s - \\sum_{m=1}^{n-1} P(A_m)$ tends to $s - s = 0$ as $n \\to \\infty$.\n5. By steps 2 to 4, $0 \\le P(L) \\le \\sum_{m \\ge n} P(A_m)$ for each $n$. Let $n \\to \\infty$ to get $P(L) = 0$.\n6. Claim: $\\omega \\in L$ if and only if $\\omega \\in A_m$ for infinitely many $m$. If $\\omega$ lies in infinitely many $A_m$, then for each $n$ some $m \\ge n$ has $\\omega \\in A_m$, so $\\omega \\in B_n$ for all $n$.\n7. Conversely, if $\\omega$ lies in only finitely many $A_m$, let $N$ exceed all these indices. Then $\\omega \\notin B_N$, so $\\omega \\notin L$.\n8. By steps 5 to 7, the set of outcomes that lie in infinitely many $A_n$ has probability $0$. So almost surely only finitely many $A_n$ occur.\n\n## Idea\n\nAn outcome that lies in infinitely many $A_n$ lies, for every $n$, in some $A_m$ with $m \\ge n$. The probability of that is at most the tail of a convergent series, which tends to $0$.\n",
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