Definition:Limit Superior/Definition 1

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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:

$\displaystyle \map {\limsup_{n \mathop \to \infty} } {x_n} = \overline l$

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.