# 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.

See Relationship between Limit Inferior and Lower Limit.

## Also see

- Definition:Limit Inferior of Sequence of Sets for an extension of this concept into the field of measure theory.

## 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.

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 5.13$ - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 8$