# 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 called the **upper limit**, or just **limsup**.

Some sources use the term **upper bound**, but that term has a wider application and is not recommended in this context.

## 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

- Definition:Limit Superior of Sequence of Sets for an extension of this concept into the field of set theory, which is important in measure theory.

- 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.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**limit superior**(of a sequence) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**limit superior**(of a sequence) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**limit superior**