Definition:Limit Superior/Definition 1

From ProofWiki
Jump to navigation Jump to search

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


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.


Sources