Chapter 5.4, Problem 25ES

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(O) is true because $r^0=1$ by definition of zeroth power.

Show that for every integer $k\ge 0$, if P(i) is True far each integer i from 0 through k, then $P\left(k+1\right)$is also true: Let k be any integer with $k\ge 0$ and suppose that $r^i=1$ for each integer i from 0 through k. This is the inductive hypothesis We must show that $r^{k+1}=1$ . Now
$r^{k+1}=r^{k+k-\left(k-1\right)}$ because $k+k-\left(k-1\right)=k+k-k+1$

$=\frac{r^k\cdot r^k}{r^{k-1}}$ by the laws of exponents
$=\frac{1\cdot 1}{1}$ by Inductive hypothesis
$=1.$

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

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