Definition:Limit Inferior of Sequence of Sets

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

$$ $$

Note that $$\liminf_{n \to\infty} E_n = \{x : x\in E_{i} \text{ for all but finitely many i}\}$$. See proof here.