Introduction to Real Analysis, 4th Edition

by Bartle, Robert G.; Sherbert, Donald R.

• ISBN13: 9780471433316
• ISBN10: 0471433314
• Edition: 4th
• Format: Hardcover
• Copyright: 2011-01-18
• Publisher: Wiley

Summary

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, analyse 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 addition 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.

Author Biography

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.

Table of Contents

Preliminaries

Sets and Functions

Mathematical Induction

Finite and Infinite Sets

The Real Numbers

The Algebraic and Order Properties of R

Absolute Value and Real Line

The Completeness Property of R

Applications of the Supremum Property

Intervals

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

Limits

Limits of Functions

Limit Theorems

Some Extensions of the Limit Concept

Continuous Functions

Continuous Functions

Combinations of Continuous Functions

Continuous Functions on Intervals

Uniform Continuity

Continuity and Gauges

Monotone and Inverse Functions

Differentiation

The Derivative

The Mean Value Theorem

L'Hospital's Rules

Taylor's Theorem

The Riemann Integral

The Riemann Integral

Riemann Integrable Functions

The Fundamental Theorem

Approximate Integration

Sequences of Functions

Pointwise and Uniform Convergence

Interchange of Limits

The Exponential and Logarithmic Functions

The Trigonometric Functions

Infinite Series

Absolute Convergence

Tests for Absolute Convergence

Tests for Nonabsolute Convergence

Series of Functions

The Generalized Riemann Integral

Definition and Main Properties

Improper and Lebesgue Integrals

Infinite Intervals

Convergence Theorems

A Glimpse into Topology

Open and Closed Sets in R

Compact Sets

Continuous Functions

Metric Spaces

Logic and Proofs

Finite and Countable Sets

The Riemann And Lebesgue Criteria

Approximate Integration

Two Examples

References

Photo Credits

Hints for Selected Exercises

Index

Table of Contents provided by Publisher. All Rights Reserved.