Section 1A

1. 3. Multiply by 1 – r.

2. 4, 5. Write the terms in the n-th expression in terms of n and put over a common denominator.

Section 1B

1. 5. Add –z to both sides of z + 0 = z + 0′.

2. 6. Add –m to both sides; show that the resulting expression for x is the desired solution.

3. 11. Multiply the identity 0 + 0 = 0 by r and use the result of Exercise 5 or 6.

4. 12. Multiply by r–1.

5. 13, 14. Adapt the divisibility argument used for r2 = 2.

Section 1C

1. 5. S + 0* = S

2. 6. For s to belong to –S, one must have r + s ∈ 0* for every r ∈ S.

3. 8. Hint: What positive rationals should the product contain? What other rationals?

Section 2A

1. 1. Convert this statement so that O5 applies.

2. 2. Start with a rational r0 < x and an irrational t0 < x, and add multiples of a sufficiently small rational.

3. 3. Can the set consisting of positive integer multiples of < be bounded?

4. 5. The set {a1, a2, …} has a least upper bound a; show that a is the desired (unique) point.

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