\begin{definition}[Definition:Limit of Sets]
Let $\Bbb S = \set {E_n : n \in \N}$ 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 $\ds \lim_{n \mathop \to \infty} E_n$, is defined as:
:$\ds \lim_{n \mathop \to \infty} E_n := \limsup_{n \mathop \to \infty} E_n$
and so also:
:$\ds \lim_{n \mathop \to \infty} E_n := \liminf_{n \mathop \to \infty} E_n$
and '''$\Bbb S$ converges to the limit'''.
\end{definition}