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 $$S$$).