= Borel-Cantelli second lemma
{c}
= Second Borel-Cantelli lemma
{c}
{synonym}
If the <events> $A_n$ are mutually <independent events> and $\sum_{n=1}^{\infty}\mathbb P(A_n)=\infty$, then $\mathbb P(\limsup_n A_n)=1$. For an infinite family, mutual <independence> means every finite subfamily satisfies the factorization defining <independent events>.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[§2.3, Theorem 2.3.7 and its proof, printed p. 70]{ref}
Proof outline: for $M\geq m$, <independent events> and $1-x\leq e^{-x}$ give
$$
\mathbb P\left(\bigcap_{n=m}^{M}A_n^c\right)
=\prod_{n=m}^{M}\left(1-\mathbb P(A_n)\right)
\leq\exp\left(-\sum_{n=m}^{M}\mathbb P(A_n)\right)\longrightarrow0.
$$
<Continuity of probability> makes each infinite tail <union> have <probability> one; their countable <intersection> has <probability> one as well.
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