Definition:Limit Inferior of a Sequence of Sets

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Definition

Suppose $\left\{{E_n : n \in \N}\right\}$ is a sequence of sets.

Then the limit inferior of the sequence, denoted $\displaystyle \liminf_{n \to \infty} \ E_n$, is defined as:

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \liminf_{n \to \infty} \ E_n$	$:=$	$\displaystyle \bigcup_{n \mathop = 0}^\infty \ \bigcap_{i \mathop = n}^\infty E_n$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({E_0 \cap E_1 \cap E_2 \cap \ldots}\right) \cup \left({E_1 \cap E_2 \cap E_3 \cap \ldots}\right) \cup \ldots$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\displaystyle \liminf_{n \to \infty} \ E_n = \left\{{x : x \in E_i \text{ for all but finitely many $i$}}\right\}$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \liminf_{n \to \infty} \ E_n\)	\(:=\)	\(\displaystyle \bigcup_{n \mathop = 0}^\infty \ \bigcap_{i \mathop = n}^\infty E_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({E_0 \cap E_1 \cap E_2 \cap \ldots}\right) \cup \left({E_1 \cap E_2 \cap E_3 \cap \ldots}\right) \cup \ldots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)