Definition:Limit Superior of Sequence of Sets

Suppose $$\{E_n : n\in\N\}$$ is a sequence of sets. Then the limit superior of the sequence, denoted $$\limsup_{n\to\infty}E_n$$, is defined as

$$ $$

Note that $$\limsup_{n \to \infty} E_n = \{x : x\in E_{i} \text{ for infinitely many i}\}$$. See proof here.