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.