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 each integer 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 each integer 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 every integer 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 each integer j with a j n.

Then use ordinary mathematical induction to show that Q(n) is true for every integer n b.

That is, prove:

1. Q(b) is true.

2. For each integer k b, if Q(k) is true then Q(k + 1) is true.