(Solution) - What is wrong with this proof Theorem For every positive integer -(2025 Original AI-Free Solution)

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What is wrong with this "proof"?
"Theorem" For every positive integer n, if x and y are positive integers with max(x, y) = n, then x = y. Basis Step: Suppose that n = 1. If max(x, y) = 1 and x and y are positive integers, we have x = 1 and y = 1.
Inductive Step: Let k be a positive integer. Assume that whenever max(x, y) = k and x and y are positive integers, then x = y. Now let max(x, y) = k + 1, where x and y are positive integers. Then max(x ? 1, y ? 1) = k, so by the inductive hypothesis, x ? 1 = y ? 1. It follows that x = y, completing the inductive step.