Introduction to Real Analysis 4th Edition by Robert Bartle, Donald Sherbert ISBN 978-0471433316 0471433316
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Product details:
ISBN 10: 0471433316
ISBN 13: 978-0471433316
Author: Robert G. Bartle, Donald R. Sherbert
This text provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations, and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with additional examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: Introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.
Table of contents:
Chapter 1: Preliminaries
Sets and Functions
Mathematical Induction
Finite and Infinite Sets
Chapter 2: The Real Numbers
The Algebraic and Order Properties of ℝ
Absolute Value and the Real Line
The Completeness Property of ℝ
Applications of the Supremum Property
Intervals
Chapter 3: Sequences and Series
Sequences and Their Limits
Limit Theorems
Monotone Sequences
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properly Divergent Sequences
Introduction to Infinite Series
Chapter 4: Limits
Limits of Functions
Limit Theorems
Some Extensions of the Limit Concept
Chapter 5: Continuous Functions
Continuous Functions
Combinations of Continuous Functions
Continuous Functions on Intervals
Uniform Continuity
Continuity and Gauges
Monotone and Inverse Functions
Chapter 6: Differentiation
The Derivative
The Mean Value Theorem
L’Hospital’s Rules
Taylor’s Theorem
Chapter 7: The Riemann Integral
Riemann Integral
Riemann Integrable Functions
The Fundamental Theorem
The Darboux Integral
Approximate Integration
Chapter 8: Sequences of Functions
Pointwise and Uniform Convergence
Interchange of Limits
The Exponential and Logarithmic Functions
The Trigonometric Functions
Chapter 9: Infinite Series
Absolute Convergence
Tests for Absolute Convergence
Tests for Nonabsolute Convergence
Series of Functions
Chapter 10: The Generalized Riemann Integral
Definition and Main Properties
Improper and Lebesgue Integrals
Infinite Intervals
Convergence Theorems
Chapter 11: A Glimpse into Topology
Open and Closed Sets in ℝ
Compact Sets
Continuous Functions
Metric Spaces
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