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?
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.