A Sequential Introduction To Real Analysis

by J Martin Speight

Overview

Real analysis provides the fundamental underpinnings for calculus, arguably the most useful and influential mathematical idea ever invented. It is a core subject in any mathematics degree, and also one which many students find challenging. A Sequential Introduction to Real Analysis gives a fresh take on real analysis by formulating all the underlying concepts in terms of convergence of sequences. The result is a coherent, mathematically rigorous, but conceptually simple development of the standard theory of differential and integral calculus ideally suited to undergraduate students learning real analysis for the first time.This book can be used as the basis of an undergraduate real analysis course, or used as further reading material to give an alternative perspective within a conventional real analysis course.

Product Details

ISBN-13:	9781783267828
Publisher:	World Scientific Publishing Europe Ltd / Imperial College Press
Publication date:	12/31/2015
Series:	Essential Textbooks In Mathematics
Pages:	276
Product dimensions:	6.20(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface vii

Foreword xi

Acknowledgements xv

Additional Resources for Instructors and Lecturers xvii

Chapter 1 Basic properties of the set of real numbers 1

1.1 Recap of set notation 1

1.2 Functions 4

1.3 Boundedness and the Axiom of Completeness 8

1.4 Some consequences of the Axiom of Completeness 12

1.5 Summary 19

1.6 Tutorial problems 20

1.7 Homework problems 20

Chapter 2 Real sequences 23

2.1 Definition and examples of real sequences 23

2.2 Convergence of a real sequence 25

2.3 Summary 31

2.4 Tutorial problems 32

2.5 Homework problems 32

Chapter 3 Limit theorems 33

3.1 Some basic limit theorems 33

3.2 The Monotone Convergence Theorem 38

3.3 Sequences and suprema 40

3.4 Summary 42

3.5 Tutorial problems 43

3.6 Homework problems 43

Chapter 4 Subsequences 45

4.1 Definition, and convergence properties 45

4.2 The Bolzano-Weierslrass Theorem 49

4.3 Summary 51

4.4 Tutorial problems 52

4.5 Homework problems 52

Chapter 5 Series 55

5.1 Definition and convergence 55

5.2 Convergence tests for series 58

5.3 Alternating series 63

5.4 Absolute convergence 66

5.5 Summary 68

5.6 Tutorial problems 69

5.7 Homework problems 69

Chapter 6 Continuous functions 71

6.1 Sequential continuity 71

6.2 Basic properties of continuous functions 73

6.3 The Intermediate Value Theorem 76

6.4 The Extreme Value Theorem 82

6.5 Summary 85

6.6 Tutorial problems 86

6.7 Homework problems 86

Chapter 7 Some symbolic logic 89

7.1 Statements and their symbolic manipulation 89

7.2 Implications 92

7.3 Quantifiers 95

7.4 Summary 103

7.5 Tutorial problems 104

7.6 Homework problems 104

Chapter 8 Limits of functions 107

8.1 The main definition 107

8.2 Limits at infinity 113

8.3 Summary 122

8.4 Tutorial problems 123

8.5 Homework problems 123

Chapter 9 Differentiable functions 125

9.1 The main definition 125

9.2 The rules of differentiation 129

9.3 Functions differentiable on an interval 134

9.4 Higher derivatives and Taylor's Theorem 142

9.5 Summary 146

9.6 Tutorial problems 147

9.7 Homework problems 147

Chapter 10 Power series 149

10.1 Definition and radius of convergence 149

10.2 Differentiability of power series 154

10.3 Properties of the exponential function 159

10.4 Elementary properties of the trigonometric functions 161

10.5 Summary 166

10.6 Tutorial problems 167

10.7 Homework problems 167

Chapter 11 Integration 169

11.1 Dissections and Riemann sums 169

11.2 Definition of the Riemann integral 172

11.3 A sequential characterization of integrability 177

11.4 Elementary properties of the Riemann integral 182

11.5 The Fundamental Theorem of the Calculus 187

11.6 Summary 192

11.7 Tutorial problems 193

11.8 Homework problems 193

Chapter 12 Logarithms and irrational powers 195

12.1 Logarithms 195

12.2 Irrational (and rational) powers 199

12.3 Summary 201

12.4 Tutorial problems 202

12.5 Homework problems 202

Chapter 13 What are the reals? 203

13.1 What are the rationals? 205

13.2 The Cauchy property 210

13.3 A sequential construction of the reals 214

Epilogue: let there be S 221

Farther reading 225

Solutions to tutorial problems 227

Index 249