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.
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