Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The Real and Complex Numbers
- 3 Real and Complex Sequences
- 4 Series
- 5 Power Series
- 6 Metric Spaces
- 7 Continuous Functions
- 8 Calculus
- 9 Some Special Functions
- 10 Lebesgue Measure on the Line
- 11 Lebesgue Integration on the Line
- 12 Function Spaces
- 13 Fourier Series
- 14 * Applications of Fourier Series
- 15 Ordinary Differential Equations
- Appendix: The Banach-Tarski Paradox
- Hints for Some Exercises
- Notation Index
- General Index
Hints for Some Exercises
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The Real and Complex Numbers
- 3 Real and Complex Sequences
- 4 Series
- 5 Power Series
- 6 Metric Spaces
- 7 Continuous Functions
- 8 Calculus
- 9 Some Special Functions
- 10 Lebesgue Measure on the Line
- 11 Lebesgue Integration on the Line
- 12 Function Spaces
- 13 Fourier Series
- 14 * Applications of Fourier Series
- 15 Ordinary Differential Equations
- Appendix: The Banach-Tarski Paradox
- Hints for Some Exercises
- Notation Index
- General Index
Summary
Section 1A
3. Multiply by 1 – r.
4, 5. Write the terms in the n-th expression in terms of n and put over a common denominator.
Section 1B
5. Add –z to both sides of z + 0 = z + 0′.
6. Add –m to both sides; show that the resulting expression for x is the desired solution.
11. Multiply the identity 0 + 0 = 0 by r and use the result of Exercise 5 or 6.
12. Multiply by r–1.
13, 14. Adapt the divisibility argument used for r2 = 2.
Section 1C
5. S + 0* = S
6. For s to belong to –S, one must have r + s ∈ 0* for every r ∈ S.
8. Hint: What positive rationals should the product contain? What other rationals?
Section 2A
1. Convert this statement so that O5 applies.
2. Start with a rational r0 < x and an irrational t0 < x, and add multiples of a sufficiently small rational.
3. Can the set consisting of positive integer multiples of < be bounded?
5. The set {a1, a2, …} has a least upper bound a; show that a is the desired (unique) point.
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- Information
- AnalysisAn Introduction, pp. 241 - 254Publisher: Cambridge University PressPrint publication year: 2004