Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 5 - Section 5.2 - Strong Induction and Well-Ordering - Exercises - Page 343: 27

Answer

Have to Show that --if the statement P(n) is true for infinitely many positive integers -n and P(n + 1) → P(n) is true -for all positive integers n, then P(n) is true for all positive integers n.

Work Step by Step

--Let:, -for a proof by contradiction, - that there is some positive integer n such that -P(n) is not true. --Let - m be the smallest positive integer greater than n for which P(m) is true; - we know that such an m exists because P(m) is true for infinitely many values of m. - But we know that P(m)→P(m−1), so P(m−1) is also true. - Thus, -(m − 1) cannot be greater than n, so (m − 1) = n and P(n) is in fact true. ---This contradiction shows that P(n) is true for all n.
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