\begin{definition}[Definition:Bounded Above Mapping]
Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$.
Then $f$ is '''bounded above on $S$''' by the upper bound $H$ {{iff}}:
:$\forall x \in S: \map f x \preceq H$
That is, {{iff}} $f \sqbrk S = \set {\map f x: x \in S}$ is bounded above by $H$.
\end{definition}