Preface

Chapter 1 Preliminaries 1

The Real Numbers 1

Sets and Functions 6

Cardinality 15

Methods of Proof 20

Chapter 2 Sequences 27

Convergence 27

Limit Theorems 35

Two-state Markov Chains 40

Cauchy Sequences 44

Supremum and Infimum 52

The Bolzano-Weierstrass Theorem 55

The Quadratic Map 60

Projects 68

Chapter 3 The Riemann Integral 73

Continuity 73

Continuous Functions on Closed Intervals 80

The Riemann Integral 87

Numerical Methods 95

Discontinuities 103

Improper Integrals 113

Projects 119

Chapter 4 Differentiation 121

Differentiable Functions 121

The Fundamental Theorem of Calculus  129

Taylor’s Theorem 134

Newton’s Method 140

Inverse Functions 147

Functions of Two Variables 151

Projects 159

Chapter 5 Sequences of Functions 163

Pointwise and Uniform Convergence 163

Limit Theorems 169

The Supremum Norm 175

Integral Equations 183

The Calculus of Variations 188

Metric Spaces 196

The Contraction Mapping Principle 203

Normed Linear Spaces  210

Projects 219

Chapter 6 Series of Functions 223

Lim sup and Lim inf 223

Series of Real Constants 228

The Weierstrass M-test 238

Power Series 245

Complex Numbers 252

Infinite Products and Prime Numbers 260

Projects 270

Chapter 7 Differential Equations 273

Local Existence 273

Global Existence 283

The Error Estimate for Euler’s Method 289

Projects 296

Chapter 8 Complex Analysis 299

Analytic Functions 299

Integration on Paths 305

Cauchy's Theorem 312

Projects 320

Chapter 9 Fourier Series 323

The Heat Equation 323

Definitions and Examples 331

Pointwise Convergence 337

Mean-square Convergence 345

Projects 355

Chapter 10 Probability Theory 359

Discrete Random Variables 359

Coding Theory 368

Continuous Random Variables 376

The Variation Metric 386

Projects 398

Bibliography 403

Symbol Index 406

Index 409