# Category:Limits Inferior

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This category contains results about Limits Inferior.

Definitions specific to this category can be found in Definitions/Limits Inferior.

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 minimum.

This minimum is called the **limit inferior**.

It can be denoted:

- $\displaystyle \map {\liminf_{n \mathop \to \infty} } {x_n} = \underline l$

### Definition 2

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

- $\displaystyle \map {\liminf_{n \mathop \to \infty} } {x_n} = \sup \set {\inf_{m \mathop \ge n} x_m: n \in \N}$

## Subcategories

This category has only the following subcategory.