Definition:Limit Superior/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 maximum.

This maximum is called the limit superior.

It can be denoted:
 * $\ds \map {\limsup_{n \mathop \to \infty} } {x_n} = \overline l$

Also see

 * Equivalence of Definitions of Limit Superior