Definition:Limit Inferior

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 lower limit, the limit inferior or just liminf.

It can be denoted $$\liminf_{n \to \infty} \left({x_n}\right) = \underline l$$.

It can be defined as:
 * $$\liminf_{n \to \infty} \left({x_n}\right) = \sup\left\{{\inf_{m \ge n} x_m: n \in \N}\right\}$$

Compare limit superior.

Also see

 * Limit Inferior of a Sequence of Sets for an extension of this concept into the field of measure theory.