\begin{definition}[Definition:Infimum of Mapping/Real-Valued Function/Definition 1]
Let $f: S \to \R$ be a real-valued function.
Let $f$ be bounded below on $S$.
The '''infimum of $f$ on $S$''' is defined by:
:$\ds \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$
where
:$\inf f \sqbrk S$ is the infimum in $\R$ of the image of $S$ under $f$.
\end{definition}