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 Superior


 * 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.