# Definition:Limit Superior

## Definition

Let $\sequence {x_n}$ be a bounded sequence in $\R$.

### Definition 1

Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$.

From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a maximum.

This maximum is called the limit superior.

It can be denoted:

$\ds \map {\limsup_{n \mathop \to \infty} } {x_n} = \overline l$

### Definition 2

The limit superior of $\sequence {x_n}$ is defined and denoted as:

$\ds \map {\limsup_{n \mathop \to \infty} } {x_n} = \inf \set {\sup_{m \mathop \ge n} x_m: n \in \N}$

## Also known as

The limit superior is also known as the upper limit, or just limsup.

## Examples

### Sequence of Reciprocals

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N_{>0}: a_n = \dfrac 1 n$

The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 0$

### Divergent Sequence $\paren {-1}^n$

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N_{>0}: a_n = \paren {-1}^n$

The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$

### Farey Sequence

Consider the Farey sequence:

$\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$

The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$

## Also see

• Results about limits superior 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.