Limit Superior includes Limit Inferior

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

Then:
 * $\displaystyle \liminf_{n \to \infty} E_n \subseteq \limsup_{n \to \infty} E_n$

Proof
Select any $\displaystyle x \in \liminf_{n \to \infty} E_n$.

Then by definition 2 of Limit Inferior of Sequence 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 definition 2 of Limit Superior of Sequence of Sets:
 * $\displaystyle x \in \limsup_{n \to \infty} E_n$