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Mathematics-Online lexicon:

supremum and infimum

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A subset $ S$ of $ \mathbb{R}$ is bounded above if there is exists a bound $ b$ 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 $ S$. 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)

see also:

  automatically generated 5/25/2009