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