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?