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 $$\forall s \in S: \exists m \in \Z: m \le s$$.

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

So:

$$ $$ $$

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

Also see

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