# proof and mathematical induction

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• Apr 11th 2007, 11:12 AM
Possible actuary
proof and mathematical induction
Given: Prove that if A is any well-ordered set of real numbers and B is a nonempty subset of A, then B is also well-ordered.

I am pretty sure that this statement is true. Do you use the Principle of Mathematical Induction to prove it?
• Apr 11th 2007, 11:16 AM
ThePerfectHacker
Quote:

Originally Posted by Possible actuary
Given: Prove that if A is any well-ordered set of real numbers and B is a nonempty subset of A, then B is also well-ordered.

I am pretty sure that this statement is true. Do you use the Principle of Mathematical Induction to prove it?

If A is a non-empty ordered set then by definition any non-trivial subset has a least element.

Let B be a non-empty subset. We will show that B is well-orded under the same relation. If C is a non-trivial subset of B then it is also a non-trivial subset of A. But A is well-ordered, hence C has a least element. Thus, B is well-ordered.