Definition:Limit Superior of Sequence of Sets

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Definition

Definition 1

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:

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


Definition 2

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:

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


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

  • Results about limits superior of set sequences can be found here.


Linguistic Note

The plural of limit superior is limits superior. This is because limit is the noun and superior is the adjective qualifying that noun.