# 22.a. The Archimedean property for the rational numbers states that for all rational numbers r, ther

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

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

Hint: If r is any rational number, let S be the set of all integers n such that r

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.