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 defined 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, if P(i) is true for all integers i from a through k, then P(k + 1) is true. The principle of strong mathematical induction would allow us to conclude immediately that P(n) is true for all integers n ≥ a. Can we reach the same conclusion using the 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, if Q(k) is true then Q(k + 1) is true.