Definition:Supremum/Mapping

Definition

Let $f$ be a mapping defined on a subset of the real numbers $S \subseteq \R$.

Let $f$ be bounded above on $S$.

It follows from the Continuum Property that the codomain of $f$ has a supremum on $S$.

Thus:

$\displaystyle \sup_{x \in S} f \left({x}\right) = \sup f \left({S}\right)$

Linguistic Note

The plural of supremum is suprema, although the (incorrect) form supremums can occasionally be found if you look hard enough.

Also see

• Infimum

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 7.13$