Definition:Limit of Sets

Suppose $$\{E_n : n \in \N\}$$ is a sequence of sets, and suppose the limit superior of the sequence equals the limit inferior of the sequence.

Then the limit of the sequence, denoted $$\lim_{n\to\infty}E_n$$, is defined as


 * $$\lim_{n\to\infty}E_n \equiv \limsup_{n\to\infty}E_n \text{ }(=\liminf_{n\to\infty}E_n)$$

and we say that the sequence converges to the limit.

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