TITLE: lim sup and lim inf of sequence of sets.
QUESTION [120 upvotes]: I was wondering if someone would be so kind to provide a very simple explanation of lim sup and lim inf of s sequence of sets. For a sequence of subsets $A_n$  of a set $X$, the $\limsup A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n \right)$ and $\liminf A_n = \bigcup_{N=1}^\infty \left(\bigcap_{n \ge N} A_n\right)$. But I am having a hard time imagining what that really means unions of intersections and intersections of unions I think maybe causing the trouble. I read the version on Wikipedia but that didn't resolve this either.
Any help would be much appreciated.

REPLY [2 votes]: Another way to think about it is via indicator functions: let $f_n$ be the indicator function of the set $X_n$ (i.e. $f_n(x)=1$ if $x\in X_n$ and 0 otherwise). Then it is not difficult to check that $\lim \sup f_n$ is the indicator function of $\lim \sup X_n$ and similarly for limit inferior.