Limit Superior includes Limit Inferior

Theorem
Let $$\{E_n : n\in\N\}$$ be a sequence of sets.

Then $$\liminf_{n\to\infty}E_n \subseteq \limsup_{n\to\infty}E_n$$.

Proof
Select any $$x\in\liminf_{n\to\infty}E_n$$. Then by the Characterization of Limit Inferior of Sets, $$x$$ belongs to $$E_i$$ for all but finitely many values of $$i$$.

Therefore $$x$$ belongs to $$E_i$$ for infinitely many values of $$i$$.

So by the Characterization of Limit Superior of Sets, $$x\in\limsup_{n\to\infty}E_n$$.