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 limit superior.

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 known as
The limit superior is also known as the upper limit, or just limsup.

Also see

 * Definition:Limit Inferior


 * Definition:Limit Superior of Sequence of Sets for an extension of this concept into the field of set theory, which is important in measure theory.

Linguistic Note
The plural of limit superior is limits superior. This is because limit is the noun and superior is the adjective qualifying that noun.