# Definition:Limit Superior

## Definition

Let $\left \langle {x_n} \right \rangle$ be a bounded sequence in $\R$.

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

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

This maximum is called the **limit superior**.

It can be denoted:

- $\displaystyle \limsup_{n \to \infty} \left({x_n}\right) = \overline l$

It can be defined as:

- $\displaystyle \limsup_{n \to \infty} \left({x_n}\right) = \inf \ \left\{{\sup_{m \ge n} x_m: n \in \N}\right\}$

## Also known as

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

## 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.

## 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

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 5.13$ - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 8$