# Book:Robert G. Bartle/Introduction to Real Analysis/Fourth Edition

## Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis (4th Edition)

Published $\text {2011}$, Wiley

ISBN 978-0-471-43331-6

### Contents

1 Preliminaries
1.1 Sets and Functions
1.2 Mathematical Induction
1.3 Finite and Infinite Sets
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
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
4 Limits
4.1 Limits of Functions
4.2 Limit Theorems
4.3 Some Extensions of the Limit Concept
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
6 Differentiation
6.1 The Derivative
6.2 The Mean Value Theorem
6.3 L’Hospital’s Rules
6.4 Taylor’s Theorem
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
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
9 Infinite Series
9.1 Absolute Convergence
9.2 Tests for Absolute Convergence
9.3 Tests for Nonabsolute Convergence
9.4 Series of Functions
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
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
Appendices
1 Logic and Proofs
2 Finite and Countable Sets
3 The Riemann and Lebesgue Criteria
4 Approximate Integration
5 Two Examples
References
Photo Credits
Hints for Selected Exercises
Index