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

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

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Lim sup and lim inf: $\S 5.13$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 1. (of a sequence)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 1. (of a sequence)