Limit of Sets Exists iff Limit Inferior contains Limit Superior

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

Then $\Bbb S$ converges to a limit iff:


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

Sufficient Condition
Let $\Bbb S$ converge to a limit.

Then by definition:
 * $\displaystyle \limsup_{n \to \infty} E_n = \liminf_{n \to \infty} E_n$

and so by definition of set equality:
 * $\displaystyle \limsup_{n \to \infty} E_n \subseteq \liminf_{n \to \infty}E_n$

Necessary Condition
Suppose that:
 * $\displaystyle \limsup_{n \to \infty} E_n \subseteq \liminf_{n \to \infty}E_n$

From Limit Superior includes Limit Inferior:
 * $\displaystyle \liminf_{n \to \infty} E_n \subseteq \limsup_{n \to \infty} E_n$

whether or not $\Bbb S$ converges to a limit.

By definition of set equality:
 * $\displaystyle \limsup_{n \to \infty} E_n = \liminf_{n \to \infty} E_n$

and so by definition $\Bbb S$ converges to a limit.