[分析] Apostol上的exercise
Prove that every nonempty set of positive integers contains a smallest memeber.
我嚐試做的證明如下:
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.
看起來好像是對的,但如果S的元素數無窮大,會不會出問題?
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