Robert Gardner Bartle was an American mathematician specializing in real analysis. He is known for writing various popular textbooks.Donald R. Sherbert is the author of Introduction to Real Analysis, 4th Edition, published by Wiley.

• CHAPTER 1 PRELIMINARIES 11.1 Sets and Functions 11.2 Mathematical Induction 121.3 Finite and Infinite Sets 16CHAPTER 2 THE REAL NUMBERS 232.1 The Algebraic and Order Properties of R 232.2 Absolute Value and the Real Line 322.3 The Completeness Property of R 362.4 Applications of the Supremum Property 402.5 Intervals 46CHAPTER 3 SEQUENCES AND SERIES 543.1 Sequences and Their Limits 553.2 Limit Theorems 633.3 Monotone Sequences 703.4 Subsequences and the Bolzano-Weierstrass Theorem 783.5 The Cauchy Criterion 853.6 Properly Divergent Sequences 913.7 Introduction to Infinite Series 94CHAPTER 4 LIMITS 1024.1 Limits of Functions 1034.2 Limit Theorems 1114.3 Some Extensions of the Limit Concept 116CHAPTER 5 CONTINUOUS FUNCTIONS 1245.1 Continuous Functions 1255.2 Combinations of Continuous Functions 1305.3 Continuous Functions on Intervals 1345.4 Uniform Continuity 1415.5 Continuity and Gauges 1495.6 Monotone and Inverse Functions 153CHAPTER 6 DIFFERENTIATION 1616.1 The Derivative 1626.2 The Mean Value Theorem 1726.3 L'Hospital's Rules 1806.4 Taylor's Theorem 188CHAPTER 7 THE RIEMANN INTEGRAL 1987.1 Riemann Integral 1997.2 Riemann Integrable Functions 2087.3 The Fundamental Theorem 2167.4 The Darboux Integral 2257.5 Approximate Integration 233CHAPTER 8 SEQUENCES OF FUNCTIONS 2418.1 Pointwise and Uniform Convergence 2418.2 Interchange of Limits 2478.3 The Exponential and Logarithmic Functions 2538.4 The Trigonometric Functions 260CHAPTER 9 INFINITE SERIES 2679.1 Absolute Convergence 2679.2 Tests for Absolute Convergence 2709.3 Tests for Nonabsolute Convergence 2779.4 Series of Functions 281CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 28810.1 Definition and Main Properties 28910.2 Improper and Lebesgue Integrals 30210.3 Infinite Intervals 30810.4 Convergence Theorems 315CHAPTER 11 A GLIMPSE INTO TOPOLOGY 32611.1 Open and Closed Sets in R 32611.2 Compact Sets 33311.3 Continuous Functions 33711.4 Metric Spaces 341APPENDIX A LOGIC AND PROOFS 348APPENDIX B FINITE AND COUNTABLE SETS 357APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 360APPENDIX D APPROXIMATE INTEGRATION 364APPENDIX E TWO EXAMPLES 367REFERENCES 370PHOTO CREDITS 371HINTS FOR SELECTED EXERCISES 372INDEX 395

Robert Gardner Bartle was an American mathematician specializing in real analysis. He is known for writing various popular textbooks.Donald R. Sherbert is the author of Introduction to Real Analysis, 4th Edition, published by Wiley.

• CHAPTER 1 PRELIMINARIES 11.1 Sets and Functions 11.2 Mathematical Induction 121.3 Finite and Infinite Sets 16CHAPTER 2 THE REAL NUMBERS 232.1 The Algebraic and Order Properties of R 232.2 Absolute Value and the Real Line 322.3 The Completeness Property of R 362.4 Applications of the Supremum Property 402.5 Intervals 46CHAPTER 3 SEQUENCES AND SERIES 543.1 Sequences and Their Limits 553.2 Limit Theorems 633.3 Monotone Sequences 703.4 Subsequences and the Bolzano-Weierstrass Theorem 783.5 The Cauchy Criterion 853.6 Properly Divergent Sequences 913.7 Introduction to Infinite Series 94CHAPTER 4 LIMITS 1024.1 Limits of Functions 1034.2 Limit Theorems 1114.3 Some Extensions of the Limit Concept 116CHAPTER 5 CONTINUOUS FUNCTIONS 1245.1 Continuous Functions 1255.2 Combinations of Continuous Functions 1305.3 Continuous Functions on Intervals 1345.4 Uniform Continuity 1415.5 Continuity and Gauges 1495.6 Monotone and Inverse Functions 153CHAPTER 6 DIFFERENTIATION 1616.1 The Derivative 1626.2 The Mean Value Theorem 1726.3 L'Hospital's Rules 1806.4 Taylor's Theorem 188CHAPTER 7 THE RIEMANN INTEGRAL 1987.1 Riemann Integral 1997.2 Riemann Integrable Functions 2087.3 The Fundamental Theorem 2167.4 The Darboux Integral 2257.5 Approximate Integration 233CHAPTER 8 SEQUENCES OF FUNCTIONS 2418.1 Pointwise and Uniform Convergence 2418.2 Interchange of Limits 2478.3 The Exponential and Logarithmic Functions 2538.4 The Trigonometric Functions 260CHAPTER 9 INFINITE SERIES 2679.1 Absolute Convergence 2679.2 Tests for Absolute Convergence 2709.3 Tests for Nonabsolute Convergence 2779.4 Series of Functions 281CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 28810.1 Definition and Main Properties 28910.2 Improper and Lebesgue Integrals 30210.3 Infinite Intervals 30810.4 Convergence Theorems 315CHAPTER 11 A GLIMPSE INTO TOPOLOGY 32611.1 Open and Closed Sets in R 32611.2 Compact Sets 33311.3 Continuous Functions 33711.4 Metric Spaces 341APPENDIX A LOGIC AND PROOFS 348APPENDIX B FINITE AND COUNTABLE SETS 357APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 360APPENDIX D APPROXIMATE INTEGRATION 364APPENDIX E TWO EXAMPLES 367REFERENCES 370PHOTO CREDITS 371HINTS FOR SELECTED EXERCISES 372INDEX 395

Robert Gardner Bartle was an American mathematician specializing in real analysis. He is known for writing various popular textbooks.Donald R. Sherbert is the author of Introduction to Real Analysis, 4th Edition, published by Wiley.

• CHAPTER 1 PRELIMINARIES 11.1 Sets and Functions 11.2 Mathematical Induction 121.3 Finite and Infinite Sets 16CHAPTER 2 THE REAL NUMBERS 232.1 The Algebraic and Order Properties of R 232.2 Absolute Value and the Real Line 322.3 The Completeness Property of R 362.4 Applications of the Supremum Property 402.5 Intervals 46CHAPTER 3 SEQUENCES AND SERIES 543.1 Sequences and Their Limits 553.2 Limit Theorems 633.3 Monotone Sequences 703.4 Subsequences and the Bolzano-Weierstrass Theorem 783.5 The Cauchy Criterion 853.6 Properly Divergent Sequences 913.7 Introduction to Infinite Series 94CHAPTER 4 LIMITS 1024.1 Limits of Functions 1034.2 Limit Theorems 1114.3 Some Extensions of the Limit Concept 116CHAPTER 5 CONTINUOUS FUNCTIONS 1245.1 Continuous Functions 1255.2 Combinations of Continuous Functions 1305.3 Continuous Functions on Intervals 1345.4 Uniform Continuity 1415.5 Continuity and Gauges 1495.6 Monotone and Inverse Functions 153CHAPTER 6 DIFFERENTIATION 1616.1 The Derivative 1626.2 The Mean Value Theorem 1726.3 L'Hospital's Rules 1806.4 Taylor's Theorem 188CHAPTER 7 THE RIEMANN INTEGRAL 1987.1 Riemann Integral 1997.2 Riemann Integrable Functions 2087.3 The Fundamental Theorem 2167.4 The Darboux Integral 2257.5 Approximate Integration 233CHAPTER 8 SEQUENCES OF FUNCTIONS 2418.1 Pointwise and Uniform Convergence 2418.2 Interchange of Limits 2478.3 The Exponential and Logarithmic Functions 2538.4 The Trigonometric Functions 260CHAPTER 9 INFINITE SERIES 2679.1 Absolute Convergence 2679.2 Tests for Absolute Convergence 2709.3 Tests for Nonabsolute Convergence 2779.4 Series of Functions 281CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 28810.1 Definition and Main Properties 28910.2 Improper and Lebesgue Integrals 30210.3 Infinite Intervals 30810.4 Convergence Theorems 315CHAPTER 11 A GLIMPSE INTO TOPOLOGY 32611.1 Open and Closed Sets in R 32611.2 Compact Sets 33311.3 Continuous Functions 33711.4 Metric Spaces 341APPENDIX A LOGIC AND PROOFS 348APPENDIX B FINITE AND COUNTABLE SETS 357APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 360APPENDIX D APPROXIMATE INTEGRATION 364APPENDIX E TWO EXAMPLES 367REFERENCES 370PHOTO CREDITS 371HINTS FOR SELECTED EXERCISES 372INDEX 395