# Definition:Limit Inferior

## 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 minimum.

This minimum is called the limit inferior.

It can be denoted:

$\displaystyle \liminf_{n \mathop \to \infty} \left({x_n}\right) = \underline l$

It can be defined as:

$\displaystyle \liminf_{n \mathop \to \infty} \left({x_n}\right) = \sup \left\{{\inf_{m \mathop \ge n} x_m: n \in \N}\right\}$

## Also known as

The limit inferior is also called the lower limit, or just liminf.

However, note that the term lower limit has a subtly different meaning in the context of topology, so to avoid ambiguity its use here is not recommended.

## Linguistic Note

The plural of limit inferior is limits inferior. This is because limit is the noun and inferior is the adjective qualifying that noun.