There is this question here where there's a definition of $\lim \sup S_n$ and $\lim \inf S_n$ where $S_n$ is a sequence of sets, specifically, a sequence of subsets of a given set.
The defintion is then given as a union of intersections, and intersection of unions, respectively.
I want to get a better handle on this definition.
I now want to construct a sequence of real intervals $S_n$ (closed or open) such that neither $\lim \sup S_n$ nor $\lim \inf S_n$ is empty, and such that they are not equal.
I keep trying but my $\lim \inf S_n$ is empty, as soon as I make my $\lim \sup S_n$ not empty and not equal to the $\lim \inf S_n$.
Thanks in advance