Definition:Limit of Sets

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

Let the limit superior of $\Bbb S$ be equal the limit inferior of $\Bbb S$.

Then the limit of $\Bbb S$, denoted $\displaystyle \lim_{n \to \infty} E_n$, is defined as:

$\displaystyle \lim_{n \to \infty} E_n := \limsup_{n \to \infty} E_n \ (=\liminf_{n \to \infty} E_n)$

and we say that $\Bbb S$ converges to the limit.

Note that because $\displaystyle \liminf_{n \to \infty} E_n \subseteq \limsup_{n \to \infty}E_n$ automatically (proof here), all that is required for $\displaystyle \lim_{n \to \infty} E_n$ to exist is for $\limsup_{n \to \infty} E_n \subseteq \liminf_{n \to \infty} E_n$.