# Category:Limits Superior

Jump to navigation
Jump to search

This category contains results about Limits Superior.

Definitions specific to this category can be found in Definitions/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:

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

## Subcategories

This category has only the following subcategory.