Use mathematical induction to prove that the statement is true for every positive integer n.If 0 < a < 1, then an < 1.(Assume that a is a constant.)

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, a1 < 1 or a < 1, which is true by assumption.
b). Assume that the statement is true for n = k:
 ak < 1
 Multiply both sides by a:
 a?ak = ak+1 < a
By assumption, a < 1, so ak+1 < 1. Since the statement is true for n = k + 1 as long as it is true for n = k, and since the statement is true for n = 1, then it is true for all natural numbers n.