# Non empty and non equal $\lim \sup S_n$ and $\lim \inf S_n$

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 $\lim \inf S_n$.

Any hints?

Thanks in advance

## 1 Answer

Similar to how oscillating functions might not have limits (say, $\sin(x)$), we can define $$S_n = \begin{cases} [0,1] &\text{if n even}\\ [-1,0] &\text{if n odd} \end{cases}$$ Then $\lim \inf S_n = \{0\}$ and $\lim \sup S_n = [-1,1]$.

• Is it always the case that $\lim \inf \subseteq \lim \sup$? – JuliusL33t Aug 3 '17 at 20:08