Find the mistake in the following proof that purports to show that every nonnegative

Chapter 5, Problem 19

(choose chapter or problem)

Find the mistake in the following “proof” that purports to show that every nonnegative integer power of every nonzero real number is 1.

“Proof: Let r be any nonzero real number and let the property P(n) be the equation \(r^{n} = 1\).

Show that P(0) is true: P(0) is true because \(r^{0} = 1\) by definition of zeroth power.

Show that for all integers k≥0, if P(i) is true for all integers i from 0 through k, then P(k + 1) is also true: Let k be any integer with k ≥ 0 and suppose that \(r^{i} = 1\) for all integers i from 0 through k. This is the inductive hypothesis. We must show that \(r^{k+1} = 1\). Now

Thus \(r^{k+1} = 1\) [as was to be shown].

[Since we have proved the basis step and the inductive step of the strong mathematical induction, we conclude that the given statement is true.]”

Text Transcription:

r^n = 1

r^0 = 1

r^i = 1

r^k+1 = 1