# Definition:Limit Inferior

## Definition

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:

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

### Definition 2

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

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

## 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.

## Examples

### Sequence of Reciprocals

Let $\sequence {a_n}$ be the sequence defined as:

- $\forall n \in \N_{>0}: a_n = \dfrac 1 n$

The limit inferior of $\sequence {a_n}$ is given by:

- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = 0$

### Divergent Sequence $\paren {-1}^n$

Let $\sequence {a_n}$ be the sequence defined as:

- $\forall n \in \N_{>0}: a_n = \paren {-1}^n$

The limit inferior of $\sequence {a_n}$ is given by:

- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = -1$

### Farey Sequence

Consider the Farey sequence:

- $\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$

The limit inferior of $\sequence {a_n}$ is given by:

- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = 0$

## Also see

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

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## 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

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**limit inferior**