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

Subject Matter

 * Real Analysis

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