A First Course In Analysis

by Donald Yau

Overview

This book is an introductory text on real analysis for undergraduate students. The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines who use real analysis. Since this book is aimed at students who do not have much prior experience with proofs, the pace is slower in earlier chapters than in later chapters. There are hundreds of exercises, and hints for some of them are included.

Product Details

ISBN-13:	9789814417853
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/22/2013
Pages:	208
Product dimensions:	6.60(w) x 9.80(h) x 0.70(d)

Table of Contents

Preface vii

1 Sets, Functions, and Real Numbers 1

1.1 Sets 1

1.2 Functions 4

1.3 Real Numbers 8

1.4 Mathematical Induction 17

1.5 Countability 21

1.6 Additional Exercises 25

2 Sequences 29

2.1 Sequences of Real Numbers 29

2.2 Properties of Limits 37

2.3 The Bolzano-Weierstrass Theorem 44

2.4 Limit Superior and Limit Inferior 51

2.5 Additional Exercises 57

3 Series 61

3.1 Convergence of Series 62

3.2 Comparison Tests 67

3.3 Alternating Series Test 70

3.4 Absolute Convergence 72

3.5 Rearrangement of Series 77

3.6 Additional Exercises 80

4 Continuous Functions 83

4.1 Limit Points 83

4.2 Limits of Functions 85

4.3 Continuity 90

4.4 Extreme and Intermediate Value Theorems 93

4.5 Uniform Continuity 96

4.6 Monotone and Inverse Functions 99

4.7 Functions of Bounded Variation 104

4.8 Additional Exercises 110

5 Differentiation 115

5.1 The Derivative 115

5.2 Mean Value Theorem 122

5.3 Taylor's Theorem 126

5.4 Additional Exercises 131

6 Integration 135

6.1 The Integral 135

6.2 Integration via Tagged Partitions 141

6.3 Basic Properties of Integrals 146

6.4 Fundamental Theorem of Calculus 149

6.5 Additional Exercises 155

7 Sequences and Series of Functions 157

7.1 Pointwise and Uniform Convergence 157

7.2 Interchange of Limits 163

7.3 Series of Functions 169

7.4 Power Series 175

7.5 Additional Exercises 181

Hints for Selected Exercises 183

List of Notations 189

Bibliography 191

Index 193