Characterizing Property of Infimum of Subset of Real Numbers

Theorem
Let $S \subset \R$ be a nonempty subset of the real numbers.

Let $S$ be bounded below.

Let $\omega \in \R$.


 * 1) $\omega$ is the infumum of $S$
 * 2) * $\omega$ is a lower bound for $S$ and
 * 3) *$\forall \epsilon>0$ there exists $x \in S$ with $\omega < x + \epsilon$
 * 1) *$\forall \epsilon>0$ there exists $x \in S$ with $\omega < x + \epsilon$

Also see

 * Characterizing Property of Supremum of Subset of Real Numbers