Definition:Infimum of Mapping/Real-Valued Function/Definition 2

Definition
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 $\displaystyle \inf_{x \mathop \in S} f \left({x}\right) := k \in \R$ such that:


 * $(1): \quad \forall x \in S: k \le f \left({x}\right)$
 * $(2): \quad \exists x \in S: \forall \epsilon \in \R_{>0}: f \left({x}\right) < k + \epsilon$

Also see

 * Equivalence of Definitions of Infimum of Real-Valued Function