This is a contradiction, because andTherefore T has a smallest element.(d) Claim. The WOP implies the PCI.Proof. Assume the WOP. To prove the PCI, let S be a subset of suchthat for all natural numbers m, Let Thenis an integer, so However,(k + 1) 1 = k, so Thus every natural number is in k S. S, so S = .k

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