Introduction to Real Analysis

CHAPTER 1 PRELIMINARIES 1

1.1 Sets and Functions 1

1.2 Mathematical Induction 12

1.3 Finite and Infinite Sets 16

CHAPTER 2 THE REAL NUMBERS 23

2.1 The Algebraic and Order Properties of R 23

2.2 Absolute Value and the Real Line 32

2.3 The Completeness Property of R 36

2.4 Applications of the Supremum Property 40

2.5 Intervals 46

CHAPTER 3 SEQUENCES AND SERIES 54

3.1 Sequences and Their Limits 55

3.2 Limit Theorems 63

3.3 Monotone Sequences 70

3.4 Subsequences and the Bolzano-Weierstrass Theorem 78

3.5 The Cauchy Criterion 85

3.6 Properly Divergent Sequences 91

3.7 Introduction to Infinite Series 94

CHAPTER 4 LIMITS 102

4.1 Limits of Functions 103

4.2 Limit Theorems 111

4.3 Some Extensions of the Limit Concept 116

CHAPTER 5 CONTINUOUS FUNCTIONS 124

5.1 Continuous Functions 125

5.2 Combinations of Continuous Functions 130

5.3 Continuous Functions on Intervals 134

5.4 Uniform Continuity 141

5.5 Continuity and Gauges 149

5.6 Monotone and Inverse Functions 153

CHAPTER 6 DIFFERENTIATION 161

6.1 The Derivative 162

6.2 The Mean Value Theorem 172

6.3 L'Hospital's Rules 180

6.4 Taylor's Theorem 188

CHAPTER 7 THE RIEMANN INTEGRAL 198

7.1 Riemann Integral 199

7.2 Riemann Integrable Functions 208

7.3 The Fundamental Theorem 216

7.4 The Darboux Integral 225

7.5 Approximate Integration 233

CHAPTER 8 SEQUENCES OF FUNCTIONS 241

8.1 Pointwise and Uniform Convergence 241

8.2 Interchange of Limits 247

8.3 The Exponential and Logarithmic Functions 253

8.4 The Trigonometric Functions 260

CHAPTER 9 INFINITE SERIES 267

9.1 Absolute Convergence 267

9.2 Tests for Absolute Convergence 270

9.3 Tests for Nonabsolute Convergence 277

9.4 Series of Functions 281

CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 288

10.1 Definition and Main Properties 289

10.2 Improper and Lebesgue Integrals 302

10.3 Infinite Intervals 308

10.4 Convergence Theorems 315

CHAPTER 11 A GLIMPSE INTO TOPOLOGY 326

11.1 Open and Closed Sets in R 326

11.2 Compact Sets 333

11.3 Continuous Functions 337

11.4 Metric Spaces 341

APPENDIX A LOGIC AND PROOFS 348

APPENDIX B FINITE AND COUNTABLE SETS 357

APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 360

APPENDIX D APPROXIMATE INTEGRATION 364

APPENDIX E TWO EXAMPLES 367

REFERENCES 370

PHOTO CREDITS 371

HINTS FOR SELECTED EXERCISES 372

INDEX 395