Definition:Limit Superior of Sequence of Sets/Definition 1

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Let $\set {E_n : n \in \N}$ be a sequence of sets.

Then the limit superior of $\set {E_n: n \in \N}$, denoted $\displaystyle \limsup_{n \mathop \to \infty} \ E_n$, is defined as:

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

Also denoted as

The limit superior of $E_n$ can also be seen denoted as:

$\displaystyle \overline {\lim}_{n \mathop \to \infty} \ E_n$

but this notation is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$ because it does not render well.

Also see