0.1 The Natural Numbers.
0.2 The Rationals.
1 The Real Numbers and Completeness.
1.1 Interval Arithmetic.
1.2 Families of Intersecting Intervals.
1.3 Fine Families.
1.4 Definition of the Reals.
1.5 Real Number Arithmetic.
1.6 Rational Approximations.
1.7 Real Intervals and Completeness.
1.8 Limits and Limiting Families.
Appendix: The Goldbach Number and Trichotomy.
2 An Inverse Function Theorem and its Application.
2.1 Functions and Inverses.
2.2 An Inverse Function Theorem.
2.3 The Exponential Function.
2.4 Natural Logs and the Euler Number.
3 Limits. Sequences and Series.
3.1 Sequences and Convergence.
3.2 Limits of Functions.
3.3 Series of Numbers.
Appendix I: Some Properties of Exp and Log.
Appendix 11: Rearrangements of Series.
4 Uniform Continuity.
4.1 Definitions and Elementary Properties.
4.2 Limits and Extensions.
Appendix I: Are there Non-Continuous Functions?
Appendix XI: Continuity of Double-Sided Inverses.
Appendix III: The Goldbach Function.
5 The Riemann Integral.
5.1 Definition and Existence.
5.2 Elementary Properties.
5.3 Extensions and Improper Integrals.
6.1 Definitions and Basic Properties.
6.2 The Arithmetic of Differentiability.
6.3 Two Important Theorems.
6.4 Derivative Tools.
6.5 Integral Tools.
7 Sequences and Series of Functions.
7.1 Sequences of Functions.
7.2 Integrals and Derivatives of Sequences.
7.3 Power Series.
7.4 Taylor Series.
7.5 The Periodic Functions.
Appendix: Binomial Issues.
8 The Complex Numbers and Fourier Series.
8.1 The Complex Numbers C.
8.2 Complex Functions and Vectors.
8.3 Fourier Series Theory.
"Very suitable for self-study by undergraduates at all levels..." (CHOICE, August 2007)
"...deserves to be read. Even if you do not subscribe to the constructive viewpoint, you'll learn something and find plenty of material to exploit in your classical analysis courses." (MAA Reviews, December 23, 2006)
- Offers an approach to introductory analysis with a constructive approach as opposed to the classical approach. There is no comparable book on the market.
- Constructivism proves a chain of results and shows, ultimately, that the quantity can be constructed. This approach is gaining appreciation as an increasingly large number of computer science and related fields are encouraging a real analysis course for students.
- Provides a unique look at the construction of real numbers as "consistent and fine families of rational intervals"
- Includes hundreds of examples throughout the book, in all ranges of difficulty and length
- Supplemented by a related web site that contains summaries of results with linked commentaries and references. Also includes links to web sites containing supplementary material and historical background.
- Authored with a friendly voice, the book encourages and helps readers to conquer difficult points.