\begin{definition}[Definition:Infimum of Mapping/Real-Valued Function/Definition 2]
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 as $\ds \inf_{x \mathop \in S} \map f x := k \in \R$ such that:
:$(1): \quad \forall x \in S: k \le \map f x$
:$(2): \quad \forall \epsilon \in \R_{>0}: \exists x \in S: \map f x < k + \epsilon$
\end{definition}