Preface ix

Introduction xi

Preliminary notation xv

1 The real numbers 1

1.1 Intuitive picture of R as points on the number line 2

1.2 The field axioms 6

1.3 Order axioms 8

1.4 The Least Upper Bound Property of R 9

1.5 Rational powers of real numbers 20

1.6 Intervals 21

1.7 Absolute value |·|and distance in R 23

1.8 (¿) Remark on the construction of R 26

1.9 Functions 28

1.10 (¿) Cardinality 40

Notes 43

2 Sequences 44

2.1 Limit of a convergent sequence 46

2.2 Bounded and monotone sequences 54

2.3 Algebra of limits 59

2.4 Sandwich theorem 64

2.5 Subsequences 68

2.6 Cauchy sequences and completeness of R 74

2.7 (¿) Pointwise versus uniform convergence 78

Notes 85

3 Continuity 86

3.1 Definition of continuity 86

3.2 Continuous functions preserve convergence 91

3.3 Intermediate Value Theorem 99

3.4 Extreme Value Theorem 106

3.5 Uniform convergence and continuity 111

3.6 Uniform continuity 111

3.7 Limits 115

Notes 124

4 Differentiation 125

4.1 Differentiable Inverse Theorem 136

4.2 The Chain Rule 140

4.3 Higher order derivatives and derivatives at boundary points 144

4.4 Equations of tangent and normal lines to a curve 148

4.5 Local minimisers and derivatives 157

4.6 Mean Value, Rolle's, Cauchy's Theorem 159

4.7 Taylor's Formula 167

4.8 Convexity 172

4.9 0/0 form of l'Hôpital's Rule 180

Notes 182

5 Integration 183

5.1 Towards a definition of the integral 183

5.2 Properties of the Riemann integral 198

5.3 Fundamental Theorem of Calculus 210

5.4 Riemann sums 226

5.5 Improper integrals 232

5.6 Elementary transcendental functions 245

5.7 Applications of Riemann Integration 278

Notes 296

6 Series 297

6.1 Series 297

6.2 Absolute convergence 305

6.3 Power series 320

Appendix 335

Notes 337

Solutions 338

Solutions to the exercises from Chapter 1 338

Solutions to the exercises from Chapter 2 353

Solutions to the exercises from Chapter 3 369

Solutions to the exercises from Chapter 4 388

Solutions to the exercises from Chapter 5 422

Solutions to the exercises from Chapter 6 475

Bibliography 493

Index 495