Definition:Supremum of Mapping/Real-Valued Function/Definition 2

Definition
Let $f$ be a real-valued function defined on a non-empty subset of the real numbers $S \subseteq \R$.

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

Let $K \in \R$.

$K$ is the supremum of $f$ iff:
 * $(1) \quad \forall x \in S: f \left({x}\right) \le K$
 * $(2) \quad \exists x \in S: \forall \epsilon \in \R_{>0}: f \left({x}\right) > K - \epsilon$

Also see

 * Equivalence of Definitions of Supremum of Real-Valued Function