Definition:Limit Inferior

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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.

Some sources use the term lower bound, but that term has a wider application and is not recommended in this context.


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


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  • Results about limits inferior can be found here.


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

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