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The Well-Ordering Principle states: The set N of positive integers is well-ordered

Chapter 4, Problem 26

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QUESTION:

The Well-Ordering Principle states: The set N of positive integers is well-ordered. (SeeExercise 25.)Prove that the Well-Ordering Principle is true if and only if the Principle of MathematicalInduction is true.

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QUESTION:

The Well-Ordering Principle states: The set N of positive integers is well-ordered. (SeeExercise 25.)Prove that the Well-Ordering Principle is true if and only if the Principle of MathematicalInduction is true.

ANSWER:

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A number  is a least element of  if  for every. For example, every finite nonempty set of real numbers and  have a least element, while  and the open interval (0, 1) of real numbers do not have a least element. A nonempty set  of real numbers is said to be well-ordered if every nonempty subset of  has a least element.

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