Let $A_{1},A_{2},\dots$ be random events in a probability space.

1. 1.

   If $\sum_{{n=1}}^{\infty}P(A_{n})<\infty$, then $P(A_{n}\operatorname{i.o.})=0$;

2. 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.