\begin{definition}[Definition:Bounded Above Mapping/Real-Valued]
Let $f: S \to \R$ be a real-valued function.
$f$ is '''bounded above on $S$''' by the upper bound $H$ {{iff}}:
:$\forall x \in S: \map f x \le H$
That is, {{iff}} the set $\set {\map f x: x \in S}$ is bounded above in $\R$ by $H$.
\end{definition}