Definition:Limit of Sets

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

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

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


 * $\displaystyle \lim_{n \to \infty} E_n := \limsup_{n \to \infty} E_n$

and so also:
 * $\displaystyle \lim_{n \to \infty} E_n := \liminf_{n \to \infty} E_n$

and $\Bbb S$ converges to the limit.

Also see

 * Limit of Sets Exists iff Limit Inferior contains Limit Superior‎: all that is required for $\displaystyle \lim_{n \to \infty} E_n$ to exist is for $\displaystyle \limsup_{n \to \infty} E_n \subseteq \liminf_{n \to \infty} E_n$.