22.a. The Archimedean property for the rational numbers states that for all rational numbers r, there is an integer n such that n > r. Prove this property. (do not solve)
22.b. Prove that given any rational number r, the number -r is also rational. (do not solve)
22.c. Use the results of parts (a) and (b) to prove that given any rational number r, there is an integer m such that m < r.
23. Use the results of exercise 22 and well-ordering principle for the integers to show that given any rational number r, there is an integer m such that m <= r < m + 1.
Hint: If r is any rational number, let S be the set of all integers n such that r < n. Use the results of exercise 22(a), 22(c), and the well-ordering principle for the integers to show that S has a least element, say v, and then show that v – 1 <= r < v.
32. Prove that if a statement can be proved by ordinary mathematical induction, then it can be proved by the well-ordering principle.
Hint: Given a predicate P(n) that satisfies conditions (1) and (2) of the principle of mathematical induction, let S be the set of all integers greater than or equal to a for which P(n) is false. Suppose that S has one or more elements, and use the well-ordering principle for the integers to derive a contradiction.
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