I did a proof, it sounds right to me, but I want to make sure.
Prove: 1 is the smallest element in Z+ (the positive integers)
Suppose not, then there exists some integer such that
Let . By our assumption that , . Thus S is not the empty set and by the Well Ordering Principle there exists a smallest element, call it . Since it follows that .
Thus, which implies that . It follows that . Thus . But , a contradiction since is the smallest element in S. Thus 1 must be the smallest element in the positive integers.
Is this right?