Prove that every nonempty set of positive integers contains a smallest memeber. 我嚐试做的证明如下: Proof: + Let the set be S⊆Z. Prove by induction. + Basis step: |S| = 1 => S = {a}, a ∈ Z. Obviously, a is the smallest memeber. Inductive step: Assume |S| = n, there is a smallest memeber b. Then, when |S| = n + 1, let c be an additional memeber. (By Axiom 6: Exactly one of the relations x = y, x < y, x > y holds.) If c ≧ b, then choose b as the smallest member. If c < b,then choose c as the smallest member. Then, there is a smallest member in S. By induction, the statement is true. 看起来好像是对的,但如果S的元素数无穷大,会不会出问题? --

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