Definition:Bounded Above Mapping/Unbounded

Definition
Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$.

Then $f$ is unbounded above on $S$ it is not bounded above on $S$:
 * $\neg \exists H \in T: \forall x \in S: \map f x \preceq H$