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# Introduction to Real Analysis

Robert G. Bartle and Donald R. Sherbert

CHAPTER 1 PRELIMINARIES.

1.1 Sets and Functions.

1.2 Mathematical Induction.

1.3 Finite and Infinite Sets.

CHAPTER 2 THE REAL NUMBERS.

2.1 The Algebraic and Order Properties of R.

2.2 Absolute Value and the Real Line.

2.3 The Completeness Property of R.

2.4 Applications of the Supremum Property.

2.5 Intervals.

CHAPTER 3 SEQUENCES AND SERIES.

3.1 Sequences and Their Limits.

3.2 Limit Theorems.

3.3 Monotone Sequences.

3.4 Subsequences and the Bolzano-Weierstrass Theorem.

3.5 The Cauchy Criterion.

3.6 Properly Divergent Sequences.

3.7 Introduction to Infinite Series.

CHAPTER 4 LIMITS.

4.1 Limits of Functions.

4.2 Limit Theorems.

4.3 Some Extensions of the Limit Concept.

CHAPTER 5 CONTINUOUS FUNCTIONS.

5.1 Continuous Functions.

5.2 Combinations of Continuous Functions.

5.3 Continuous Functions on Intervals.

5.4 Uniform Continuity.

5.5 Continuity and Gauges.

5.6 Monotone and Inverse Functions.

CHAPTER 6 DIFFERENTIATION.

6.1 The Derivative.

6.2 The Mean Value Theorem.

6.3 L’Hospital’s Rules.

6.4 Taylor’s Theorem.

CHAPTER 7 THE RIEMANN INTEGRAL.

7.1 Riemann Integral.

7.2 Riemann Integrable Functions.

7.3 The Fundamental Theorem.

7.4 The Darboux Integral.

7.5 Approximate Integration.

CHAPTER 8 SEQUENCES OF FUNCTIONS.

8.1 Pointwise and Uniform Convergence.

8.2 Interchange of Limits.

8.3 The Exponential and Logarithmic Functions.

8.4 The Trigonometric Functions.

CHAPTER 9 INFINITE SERIES.

9.1 Absolute Convergence.

9.2 Tests for Absolute Convergence.

9.3 Tests for Nonabsolute Convergence.

9.4 Series of Functions.

CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL.

10.1 Definition and Main Properties.

10.2 Improper and Lebesgue Integrals.

10.3 Infinite Intervals.

10.4 Convergence Theorems.

CHAPTER 11 A GLIMPSE INTO TOPOLOGY.

11.1 Open and Closed Sets in R.

11.2 Compact Sets.

11.3 Continuous Functions.

11.4 Metric Spaces.

APPENDIX A LOGIC AND PROOFS.

APPENDIX B FINITE AND COUNTABLE SETS.

APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA.

APPENDIX D APPROXIMATE INTEGRATION.

APPENDIX E TWO EXAMPLES.

REFERENCES.

PHOTO CREDITS.

HINTS FOR SELECTED EXERCISES.

INDEX.