Real Analysis: A Constructive Approach

Preface

Acknowledgements

Introduction

0 Preliminaries

0.1 The Natural Numbers

0.2 The Rationals

1 The Real Numbers and Completeness

1.0 Introduction

1.2 Interval Arithmetic

1.3Fine Families

1.4Definition 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

2.0 Introduction

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

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 II: 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 II: 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 Differentiation

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 and Functions

7.2 Integrals and Derivatives o Sequences

7.3 Power Series

7.4 Taylor Series

7.5 The Periodic Functions

Appendix : Binomial Issues

8 The Complex Numbers and Fourier Series

8.0 Introduction

8.1 The Complex Numbers C

8.2 Complex Functions and Vectors

8.3 Fourier Series Theory

References

Index