# Definition:Limit Superior of Sequence of Sets

## 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:

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

### 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.