# Definition:Limit Superior of Sequence of Sets

## Contents

## 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.