Set of Integers Bounded Below by Integer has Smallest Element

Theorem

Let $\Z$ be the set of integers.

Let $\O \subset S \subseteq \Z$ such that $S$ is bounded below in $\struct {\Z, \le}$.

We have that $S$ is bounded below in $\Z$.

So:

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

Hence:

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

Thus:

$T = \set {s - m: s \in S} \subseteq \N$

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

Hence:

$\paren {\forall s \in S: b_T \le s - m} \land \paren {\exists b_S \in S: b_T = b_S - m}$

So:

$\ds $

$\ds s \in S: b_S - m \le s - m$

$\ds $

$\leadsto$

$\ds \forall s \in S: b_S \le s$

Cancellability of elements of $\Z$

$\ds $

$\leadsto$

$\ds b_S \in S \land \paren {\forall s \in S: b_S \le s}$

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

$\blacksquare$