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 rn = 1.
Show that P(0) is true: P (0) is true because r0 = 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 ri = 1 for all integers i from 0 through k. This is the inductive hypothesis. We must show that rk+1 = 1. Now
Thus rk+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.]”
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