Solved: Prove that if a statement can be proved by strong

Chapter 5, Problem 5.192

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Prove that if a statement can be proved by strong mathematical induction, then it can be proved by ordinary mathematical induction. To do this, let P(n) be a property that is dened for integers n, and suppose the following two statements are true: 1. P(a), P(a+1), P(a+2),...,P(b). 2. For any integer k b, ifP(i) is true for all integers i from a through k, thenP(k+1) is true. The principle of strong mathematical induction would allow us to conclude immediately that P(n) is true for all integersn a.Canwereachthesameconclusionusingthe principle of ordinary mathematical induction? Yes! To see this, let Q(n) be the property P(j) is true for all integers j with a j n. Then use ordinary mathematical induction to show that Q(n) is true for all integers n b. That is, prove 1. Q(b) is true. 2. For any integer k b, ifQ(k) is true then Q(k+1) is true.

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