Definition:Bounded Above

Ordered Set
Let $\left({S, \preceq}\right)$ be a poset.

A subset $T \subseteq S$ is bounded above (in $S$) if:
 * $\exists M \in S: \forall a \in T: a \preceq M$

That is, there is an element of $S$ (at least one) that succeeds all the elements in $T$.

If there is no such element, then $T$ is unbounded above (in $S$).

Mapping
Let $f$ be a mapping defined on a poset $\left({S, \preceq}\right)$.

Then $f$ is said to be bounded above (in $S$) by the upper bound $H$ iff:
 * $\forall x \in S: f \left({x}\right) \preceq H$.

That is, iff $f \left({S}\right) = \left\{{f \left({x}\right): x \in S}\right\}$ is bounded above by $H$.

If there is no such $H \in S$ then $f$ is unbounded above (in $S$).

Also see

 * Upper bound
 * Bounded below
 * Lower bound
 * Bounded

Sources for Ordered Set

 * : $\S 3$
 * : $\S 14$
 * : $\S 1.1$
 * : $\S 2.2$
 * : $\S 10$

Sources for Mapping

 * : $\S 7.13$