Real analysis: a constructive approach

Abstract

Preface. Acknowledgements. Introduction. 0 Preliminaries. 0.1 The Natural Numbers. 0.2 The Rationals. 1 The Real Numbers and Completeness. 1.0 Introduction. 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. 2.0 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 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 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 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. 8.0 Introduction. 8.1 The Complex Numbers C. 8.2 Complex Functions and Vectors. 8.3 Fourier Series Theory. References. Index.