# Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.6

6. Show that every nonempty set of negative integers has a greatest element.

Solution: We assume that S is a non-empty set of negative integers.

We now consider the set
$T=\left\{-s:s\in S\right\}$

Now, T is a non-empty set of positive integers and by the well ordering principle has a least element $-s_0$ for some $s_0 \in S$

Then,
$-s_0\leq-s$
For every $s\in S$

Hence,
$s_0\geq s$
For every $s\in S$

This implies that $s_0$ is the greatest element of S.

Therefore, every non-empty set of negative integers has a greatest element.