Let $A_1,A_2,\dots$ be a sequence of sets. The limit superior of sets is defined by $$\limsup A_n = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k.$$

It is easy to see that $x\in \limsup A_n$ if and only if $x\in A_n$ for infinitely many values of $n$ Because of this, in probability theory the notation $[A_n \operatorname{i.o.}]$ is often used to refer to $\limsup A_n$ where i.o. stands for infinitely often.

The limit inferior of sets is defined by

$$\liminf A_n = \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty A_k,$$

and it can be shown that $x\in \liminf A_n$ if and only if $x$ belongs to $A_n$ for all but finitely many values of $n$