Using the well ordering principle to prove a certain property of an integer

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

By contradiction

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