Definition:Limit Superior

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 maximum.

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

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

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

Also see

 * Limit inferior


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