limit superior of sets

Let $A_1,A_2,\mathrm{\dots }$ be a sequence of sets.The limit superior of sets is defined by

$$lim\; sup{A}_n=\bigcap _{n=1}^{\mathrm{\infty }}\bigcup _{k=n}^{\mathrm{\infty }}{A}_k.$$

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

The limit inferior of sets is defined by

$$lim\; inf{A}_n=\bigcup _{n=1}^{\mathrm{\infty }}\bigcap _{k=n}^{\mathrm{\infty }}{A}_k,$$

and it can be shown that $x\in lim\; inf{A}_n$ if and only if $x$ belongs to $A_n$ for all but finitely many values of $n$.