Borel-Cantelli lemma

Let $A_1, A_2,\dots$ be random events in a probability space.

1. If $\sum_{n=1}^\infty P(A_n)<\infty$ , then $P(A_n \operatorname{i.o.}) = 0$ ;
2. If $A_1,A_2,\dots$ are independent, and $\sum_{n=1}^\infty P(A_n)=\infty$ , then $P(A_n \operatorname{i.o.})=1$

where $A=[A_n \operatorname{i.o.}]$ represents the event ``$A_n$ happens for infinitely many values of $n$ .'' Formally, $A = \limsup A_n$ , which is a limit superior of sets.