Definition:Limit Inferior/Definition 1

Definition
Let $\sequence {x_n}$ be a bounded sequence in $\R$. 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:
 * $\ds \map {\liminf_{n \mathop \to \infty} } {x_n} = \underline l$

Also see

 * Equivalence of Definitions of Limit Inferior