Mathematics-Online lexicon: supremum and infimum

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	Mathematics-Online lexicon:
	supremum and infimum

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overview

A subset

of $\mathbb{R}$ is bounded above if there is exists a bound

such that

$\displaystyle x\le b \quad \forall x\in S\, .$

The completeness axiom in $\mathbb{R}$ guarantees the existence of a least upper bound, denoted by

$\displaystyle \sup S = \sup_{x\in S} x$

Note that the supremum does not necessarily have to belong to . Thus, sets that are bounded above are not required to contain a maximal element $\max S = \max_{x\in S} x$ .

Analogously lower bounds are defined; the greatest lower bound is denoted by $\inf S$ , and a minimal element is denoted by $\min S$ .

(Authors: Höllig/Kopf/Abele)

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automatically generated 5/25/2009