Definition:Limit Superior

Let $$\left \langle {x_n} \right \rangle$$ be a bounded sequence in $\mathbb{R}$.

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 or just limsup.

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

Compare limit inferior.