The Well ordering principle states that
A least element exists in every non empty set of positive integers
Use the well Ordering principle to prove the following statement
' Any nonempty subset of negative integers has the greatest element '.
What I tried
I assume the statement
Any nonempty subset of negative integers has the least element to be true.
This means that the least element $a$ must be a negative integer but this contradicts with the Well ordering principle which states that the least element must be a positive integer. Hence the least element cannot be positive and negative at the same time which thus proves the orginal statement. Is my proof correct. Could anyone explain. Thanks