**Pure and Applied Undergraduate Texts**

Volume: 38;
2019;
302 pp;
Hardcover

MSC: Primary 03; 97; 26;

**Print ISBN: 978-1-4704-5144-8
Product Code: AMSTEXT/38**

List Price: $82.00

AMS Member Price: $65.60

MAA Member Price: $73.80

**Electronic ISBN: 978-1-4704-5289-6
Product Code: AMSTEXT/38.E**

List Price: $82.00

AMS Member Price: $65.60

MAA Member Price: $73.80

#### Supplemental Materials

# Real Analysis: A Constructive Approach Through Interval Arithmetic

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*Mark Bridger*

Real Analysis: A Constructive Approach Through Interval
Arithmetic presents a careful treatment of calculus and its
theoretical underpinnings from the constructivist point of view. This
leads to an important and unique feature of this book: All existence
proofs are direct, so showing that the numbers or functions in
question exist means exactly that they can be explicitly
calculated. For example, at the very beginning, the real numbers are
shown to exist because they are constructed from the rationals using
interval arithmetic. This approach, with its clear analogy to
scientific measurement with tolerances, is taken throughout the book
and makes the subject especially relevant and appealing to students
with an interest in computing, applied mathematics, the sciences, and
engineering.

The first part of the book contains all the usual material in a
standard one-semester course in analysis of functions of a single real
variable: continuity (uniform, not pointwise), derivatives, integrals,
and convergence. The second part contains enough more technical
material—including an introduction to complex variables and
Fourier series—to fill out a full-year course. Throughout the
book the emphasis on rigorous and direct proofs is supported by an
abundance of examples, exercises, and projects—many with
hints—at the end of every section. The exposition is informal
but exceptionally clear and well motivated throughout.

#### Readership

Undergraduate and graduate students interested in real analysis, constructivism, and logic.

#### Reviews & Endorsements

Here is another new textbook for undergraduate analysis, but it's far from a traditional approach. This one is dedicated to being entirely constructive. When we see constructivist approaches, we tend to think that they involve a lot of work to get basically the same thing, or even a bit less. This book might possibly convince you of their value.

-- Bill Satzer, MAA Reviews

#### Table of Contents

# Table of Contents

## Real Analysis: A Constructive Approach Through Interval Arithmetic

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Acknowledgments xi12
- Introduction xiii14
- Chapter 0. Preliminaries 118
- Chapter 1. The Real Numbers and Completeness 1128
- 1.0. Introduction 1128
- 1.1. Interval Arithmetic 1229
- 1.2. Families of Intersecting Intervals 2239
- 1.3. Fine Families 3249
- 1.4. Definition of the Reals 3956
- 1.5. Real Number Arithmetic 4360
- 1.6. Rational Approximations 5572
- 1.7. Real Intervals and Completeness 5875
- 1.8. Limits and Limiting Families 6380
- Appendix: The Goldbach Number and Trichotomy 6784

- Chapter 2. An Inverse Function Theorem and Its Applications 6986
- Chapter 3. Limits, Sequences, and Series 99116
- Chapter 4. Uniform Continuity 139156
- Chapter 5. The Riemann Integral 165182
- Chapter 6. Differentiation 185202
- Chapter 7. Sequences and Series of Functions 223240
- Chapter 8. The Complex Numbers and Fourier Series 271288
- References 295312
- Index 297314