# Definition:Limit Superior of Sequence of Sets

## Definition

### Definition 1

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

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

\(\ds \limsup_{n \mathop \to \infty} E_n\) | \(:=\) | \(\ds \bigcap_{i \mathop = 0}^\infty \bigcup_{n \mathop = i}^\infty E_n\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {E_0 \cup E_1 \cup E_2 \cup \ldots} \cap \paren {E_1 \cup E_2 \cup E_3 \cup \ldots} \cap \ldots\) |

### Definition 2

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

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

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

## Also denoted as

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

- $\ds \underset {n \mathop \to \infty} {\overline \lim} E_n$

but this notation is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources merely present this as:

- $\ds \overline \lim E_n$

The abbreviated notation $E^*$ can also be seen.

## Also known as

The **limit superior** of a sequence of sets is also known as its **superior limit**.

## Also see

- Definition:Limit Inferior of Sequence of Sets
- Definition:Limit of Sets
- Definition:Infinitely Often (Probability Theory)

- 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.