\begin{definition}[Definition:Limit Superior/Definition 1]
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:
:$\ds \map {\limsup_{n \mathop \to \infty} } {x_n} = \overline l$
\end{definition}