Definition:Limit Superior 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 superior of the sequence, denoted $\displaystyle \limsup_{n\to\infty} \ E_n$, is defined as:

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

$\displaystyle \limsup_{n \to \infty} \ E_n = \left\{{x : x \in E_i \text{ for infinitely many i}}\right\}$

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