# Definition:Limit Superior/Definition 1

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## Contents

## Definition

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

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:

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

## Also see

## 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): $\S 5$: Subsequences: Lim sup and lim inf: $\S 5.13$