Set of Integers Bounded Below by Integer has Smallest Element

Theorem
Let $\Z$ be the set of integers.

Let $\varnothing \subset S \subseteq \Z$ such that $S$ is bounded below.

Then $S$ has a minimal or "smallest" element.

Proof
$S$ is bounded below, so $\exists m \in \Z: \forall s \in S: m \le s$.

Hence $\forall s \in S: 0 \le s - m$.

Thus the set $T = \left\{{s - m: s \in S}\right\} \subseteq \N$.

The Well-Ordering Principle gives that $T$ has a minimal element, which we can call $b_T \in T$.

Hence $\left({\forall s \in S: b_T \le s - m}\right) \land \left({\exists b_S \in S: b_T = b_S - m}\right)$.

So:

So $b_S$ is the minimal element of $S$.

Also see

 * Integers Bounded Above has Maximal Element
 * Well-Ordering Principle