Definition:Limit Superior

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 maximum.

This maximum is called the upper limit, the limit superior or just limsup.

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

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

Compare limit inferior.

Also see

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