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I'm teaching out of Rosen's discrete math book, and for mathematical induction he says:
"Why is mathematical induction a valid proof technique? The reason comes from the well-ordering property, listed in Appendix 1, as an axiom for the set of positive integers, which states that every nonempty subset of the set of positive integers has a least element. So, suppose we know that P(1) is true and that the proposition P(k) → P(k + 1) is true for all positive integers k. To show that P(n) must be true for all positive integers n, assume that there is at least one positive integer for which P(n) is false. Then the set S of positive integers for which P(n) is false is nonempty. Thus, by the well-ordering property, S has a least element, which will be denoted by m. We know that m cannot be 1, because P(1) is true. Because m is positive and greater than 1, m − 1 is a positive integer. Furthermore, because m − 1 is less than m, it is not in S, so P(m − 1) must be true. Because the conditional statement P(m − 1) → P(m) is also true, it must be the case that P(m) is true. This contradicts the choice of m. Hence, P(n) must be true for every positive integer n."
Everything makes sense except for this sentence:
"Furthermore, because m − 1 is less than m, it is not in S..."
That doesn't appear to be part of the well-ordering property, or anything else I could find in the book. Where did that come from and how can it be explained? I see no logical reason m-1 cannot be in S.