Question: What is wrong with this proof by strong induction? Theorem For every nonnegative integer n, 5n = 0. Basis Step: 5 0 = 0.

What is wrong with this "proof" by strong induction?
"Theorem" For every nonnegative integer n, 5n = 0.
Basis Step: 5 · 0 = 0.
Inductive Step: Suppose that 5j = 0 for all nonnegative integers j with 0 ≤ j ≤ k. Write k + 1 = i + j , where i and j are natural numbers less than k + 1. By the inductive hypothesis, 5(k + 1) = 5(i + j) = 5i + 5j = 0 + 0 = 0.

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