# Category:Definitions/Limits Superior

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This category contains definitions related to Limits Superior.

Related results can be found in Category:Limits Superior.

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:

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

### Definition 2

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

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

## Pages in category "Definitions/Limits Superior"

The following 3 pages are in this category, out of 3 total.