= Borel-Cantelli first lemma
{c}
= First Borel-Cantelli lemma
{c}
{synonym}
For arbitrary <events> $A_n$ in one <probability space>,
$$
\sum_{n=1}^{\infty}\mathbb P(A_n)<\infty
\quad\Longrightarrow\quad
\mathbb P\left(\limsup_{n\to\infty}A_n\right)=0.
$$
Equivalently, only finitely many <events> occur <almost surely>; no <independence> is needed.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[§2.3, Theorem 2.3.1, printed p. 68]{ref}
Proof outline: put $B_m=\bigcup_{n\geq m}A_n$. The <union bound> gives
$$
0\leq\mathbb P\left(\limsup_n A_n\right)
\leq\mathbb P(B_m)\leq\sum_{n=m}^{\infty}\mathbb P(A_n)\longrightarrow0.
$$
The tail of a convergent nonnegative <series> tends to zero.
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