I'm getting a little confused about the logic behind a proof in a book. It is proving the well-ordering property using induction; it proves the contrapositive:
If T is a subset of the natural numbers which has no smallest element, then T is empty
Suppose T has no smallest element and let S be the complement of T. (*) Let n be a natural number such that for all m<n, m is in S. Then n is in S and so S is the set of natural numbers and T is empty.
(*) This is the step I don't understand. What if 1 is in T? Then no such n exists and S is empty? Where is the step that uses the fact T has no smallest element?