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.

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