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:

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


Definition 2

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

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


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:

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


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:

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


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:

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


Also see



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.