Integers Bounded Above has Greatest Element

Theorem

Let $\Z$ be the set of integers.

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

Then $S$ has a greatest element.

Proof

$S$ is bounded above, so $\exists M \in \Z: \forall s \in S: s \le M$.

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

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

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

Hence:

$\left({\forall s \in S: b_T \le M - s}\right) \land \left({\exists g_S \in S: b_T = M - g_S}\right)$

So:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \forall s \in S: M - g_S \le M - s\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \forall s \in S: -g_S \le -s\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Cancellability of elements of $\Z$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \forall s \in S: g_S \ge s\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle g_S \in S \land \left({\forall s \in S: g_S \ge s}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	... which is how the greatest element is defined.

So $g_S$ is the greatest element of $S$.

$\blacksquare$

Alternative Proof

Since $S$ is bounded above, $\exists M \in \Z: \forall s \in S: s \le M$.

Hence we can define the set $S\,^\prime = \left\{{-s: s \in S}\right\}$.

$S\,^\prime$ is bounded below by $-M$.

So from Integers Bounded Below has Smallest Element, $S\,^\prime$ has a smallest element, $-g_S$, say, where $\forall s \in S: -g_S \le -s$.

Therefore $g_S \in S$ (by definition of $S\,^\prime$) and $\forall s \in S: s \le g_S$.

So $g_S$ is the greatest element of $S$.

$\blacksquare$

Also see

• Integers Bounded Below has Smallest Element
• Well-Ordering Principle

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 3.6 \ (3)$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 10.2$