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by Brien S. Thomson, Judith B. Bruckner and Andrew M. Bruckner

Edition: 01Copyright: 2001

Publisher: Prentice Hall, Inc.

Published: 2001

International: No

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For undergraduate courses in Advanced Calculus and Real Analysis, as well as for beginning graduate students in mathematics. For a one or two semester sequence.

Elementary Real Analysis is written in a reader friendly style with motivational and historical material that emphasizes the ''big picture'' and makes proofs seem natural rather than mysterious. It introduces key concepts such as point set theory, uniform continuity of functions and uniform convergence of sequences of functions. It is designed to prepare students for graduate work in mathematical analysis.

**Thomson, Brian S. : Simon Frasier University **

**Bruckner, Andrew M. : University of California-Santa Barbara**

1. Properties of the Real Numbers

Introduction

The Real Number System

Algebraic Structure

Order Structure

Bounds

Sups and Infs

The Archimedean Property

Inductive Property of IN

The Rational Numbers Are Dense

The Metric Structure of R

Challenging Problems for Chapter 1

2. Sequences

Introduction

Sequences

Countable Sets

Convergence

Divergence

Boundedness Properties of Limits

Algebra of Limits

Order Properties of Limits

Monotone Convergence Criterion

Examples of Limits

Subsequences

Cauchy Convergence Criterion

Upper and Lower Limits

Challenging Problems for Chapter 2

3. Infinite Sums

Introduction

Finite Sums

Infinite Unordered Sums

Ordered Sums: Series

Criteria for Convergence

Tests for Convergence

Rearrangements

Products of Series

Summability Methods

More on Infinite Sums

Infinite Products

Challenging Problems for Chapter 3

4. Sets of Real Numbers

Introduction

Points

Sets

Elementary Topology

Compactness Arguments

Countable Sets

Challenging Problems for Chapter 4

5. Continuous Functions

Introduction to Limits

Properties of Limits

Limits Superior and Inferior

Continuity

Properties of Continuous Functions

Uniform Continuity

Extremal Properties

Darboux Property

Points of Discontinuity

Challenging Problems for Chapter 5

6. More on Continuous Functions and Sets

Introduction

Dense Sets

Nowhere Dense Sets

The Baire Category Theorem

Cantor Sets

Borel Sets

Oscillation and Continuity

Sets of Measure Zero

Challenging Problems for Chapter 6

7. Differentiation

Introduction

The Derivative

Computations of Derivatives

Continuity of the Derivative?

Local Extrema

Mean Value Theorem

Monotonicity Dini Derivatives

The Darboux Property of the Derivative

Convexity

L'Hopital's Rule

Taylor Polynomials

Challenging Problems for Chapter 7

8. The Integral

Introduction

Cauchy's First Method

Properties of the Integral

Cauchy's Second Method

Cauchy's Second Method (Continued)

The Riemann Integral

Properties of the Riemann Integral

The Improper Riemann Integral

More on the Fundamental Theorem of Calculus

Challenging Problems for Chapter 8

9. Sequences and Series of Functions

Introduction

Pointwise Limits

Uniform Limits

Uniform Convergence and Continuity

Uniform Convergence and the Integral

Uniform Convergence and Derivatives

Pompeiu's Function

Continuity and Pointwise Limits

Challenging Problems for Chapter 9

10. Power Series

Introduction Power Series: Convergence

Uniform Covergence

Functions Represented by Power Series

The Taylor Series

Products of Power Series

Composition of Power Series

Trigonometric Series

11. The Euclidean Spaces Rn

The Algebraic Structure of Rn

The Metric Structure of Rn

Elementary Topology of Rn

Sequences in Rn

Functions and Mappings

Limits of Functions from Rn to Rm

Continuity of Functions from Rn to Rm

Compact Sets in Rn

Continuous Functions on Compact Sets

Additional Remarks

12. Differentiation on Rn

Introduction Partial and Directional Derivatives

Integrals Depending on a Parameter

Differentiable Functions

Chain Rules

Implicit Function Theorems

Functions from R to Rm

Functions from Rn to Rm

13. Metric Spaces

Introduction

Metric Spaces--Specific Examples

Convergence

Sets in a Metric Space

Functions

Separable Spaces

Complete Spaces

Contraction Maps

Applications of Contraction Maps (I)

Applications of Contraction Maps (II)

Compactness

Baire Category Theorem

Applications of the Baire Category Theorem

Challenging Problems for Chapter 13

Appendix A: Background

Should I Read This Chapter?

Notation

What Is Analysis?

Why Proofs?

Indirect Proof

Contraposition

Counterexamples

Induction Quantifiers

Appendix B:

Hints for Selected Exercises

Subject Index

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Summary

For undergraduate courses in Advanced Calculus and Real Analysis, as well as for beginning graduate students in mathematics. For a one or two semester sequence.

Elementary Real Analysis is written in a reader friendly style with motivational and historical material that emphasizes the ''big picture'' and makes proofs seem natural rather than mysterious. It introduces key concepts such as point set theory, uniform continuity of functions and uniform convergence of sequences of functions. It is designed to prepare students for graduate work in mathematical analysis.

Author Bio

**Thomson, Brian S. : Simon Frasier University **

**Bruckner, Andrew M. : University of California-Santa Barbara**

Table of Contents

1. Properties of the Real Numbers

Introduction

The Real Number System

Algebraic Structure

Order Structure

Bounds

Sups and Infs

The Archimedean Property

Inductive Property of IN

The Rational Numbers Are Dense

The Metric Structure of R

Challenging Problems for Chapter 1

2. Sequences

Introduction

Sequences

Countable Sets

Convergence

Divergence

Boundedness Properties of Limits

Algebra of Limits

Order Properties of Limits

Monotone Convergence Criterion

Examples of Limits

Subsequences

Cauchy Convergence Criterion

Upper and Lower Limits

Challenging Problems for Chapter 2

3. Infinite Sums

Introduction

Finite Sums

Infinite Unordered Sums

Ordered Sums: Series

Criteria for Convergence

Tests for Convergence

Rearrangements

Products of Series

Summability Methods

More on Infinite Sums

Infinite Products

Challenging Problems for Chapter 3

4. Sets of Real Numbers

Introduction

Points

Sets

Elementary Topology

Compactness Arguments

Countable Sets

Challenging Problems for Chapter 4

5. Continuous Functions

Introduction to Limits

Properties of Limits

Limits Superior and Inferior

Continuity

Properties of Continuous Functions

Uniform Continuity

Extremal Properties

Darboux Property

Points of Discontinuity

Challenging Problems for Chapter 5

6. More on Continuous Functions and Sets

Introduction

Dense Sets

Nowhere Dense Sets

The Baire Category Theorem

Cantor Sets

Borel Sets

Oscillation and Continuity

Sets of Measure Zero

Challenging Problems for Chapter 6

7. Differentiation

Introduction

The Derivative

Computations of Derivatives

Continuity of the Derivative?

Local Extrema

Mean Value Theorem

Monotonicity Dini Derivatives

The Darboux Property of the Derivative

Convexity

L'Hopital's Rule

Taylor Polynomials

Challenging Problems for Chapter 7

8. The Integral

Introduction

Cauchy's First Method

Properties of the Integral

Cauchy's Second Method

Cauchy's Second Method (Continued)

The Riemann Integral

Properties of the Riemann Integral

The Improper Riemann Integral

More on the Fundamental Theorem of Calculus

Challenging Problems for Chapter 8

9. Sequences and Series of Functions

Introduction

Pointwise Limits

Uniform Limits

Uniform Convergence and Continuity

Uniform Convergence and the Integral

Uniform Convergence and Derivatives

Pompeiu's Function

Continuity and Pointwise Limits

Challenging Problems for Chapter 9

10. Power Series

Introduction Power Series: Convergence

Uniform Covergence

Functions Represented by Power Series

The Taylor Series

Products of Power Series

Composition of Power Series

Trigonometric Series

11. The Euclidean Spaces Rn

The Algebraic Structure of Rn

The Metric Structure of Rn

Elementary Topology of Rn

Sequences in Rn

Functions and Mappings

Limits of Functions from Rn to Rm

Continuity of Functions from Rn to Rm

Compact Sets in Rn

Continuous Functions on Compact Sets

Additional Remarks

12. Differentiation on Rn

Introduction Partial and Directional Derivatives

Integrals Depending on a Parameter

Differentiable Functions

Chain Rules

Implicit Function Theorems

Functions from R to Rm

Functions from Rn to Rm

13. Metric Spaces

Introduction

Metric Spaces--Specific Examples

Convergence

Sets in a Metric Space

Functions

Separable Spaces

Complete Spaces

Contraction Maps

Applications of Contraction Maps (I)

Applications of Contraction Maps (II)

Compactness

Baire Category Theorem

Applications of the Baire Category Theorem

Challenging Problems for Chapter 13

Appendix A: Background

Should I Read This Chapter?

Notation

What Is Analysis?

Why Proofs?

Indirect Proof

Contraposition

Counterexamples

Induction Quantifiers

Appendix B:

Hints for Selected Exercises

Subject Index

Publisher Info

Publisher: Prentice Hall, Inc.

Published: 2001

International: No

Published: 2001

International: No

Elementary Real Analysis is written in a reader friendly style with motivational and historical material that emphasizes the ''big picture'' and makes proofs seem natural rather than mysterious. It introduces key concepts such as point set theory, uniform continuity of functions and uniform convergence of sequences of functions. It is designed to prepare students for graduate work in mathematical analysis.

**Thomson, Brian S. : Simon Frasier University **

**Bruckner, Andrew M. : University of California-Santa Barbara**

1. Properties of the Real Numbers

Introduction

The Real Number System

Algebraic Structure

Order Structure

Bounds

Sups and Infs

The Archimedean Property

Inductive Property of IN

The Rational Numbers Are Dense

The Metric Structure of R

Challenging Problems for Chapter 1

2. Sequences

Introduction

Sequences

Countable Sets

Convergence

Divergence

Boundedness Properties of Limits

Algebra of Limits

Order Properties of Limits

Monotone Convergence Criterion

Examples of Limits

Subsequences

Cauchy Convergence Criterion

Upper and Lower Limits

Challenging Problems for Chapter 2

3. Infinite Sums

Introduction

Finite Sums

Infinite Unordered Sums

Ordered Sums: Series

Criteria for Convergence

Tests for Convergence

Rearrangements

Products of Series

Summability Methods

More on Infinite Sums

Infinite Products

Challenging Problems for Chapter 3

4. Sets of Real Numbers

Introduction

Points

Sets

Elementary Topology

Compactness Arguments

Countable Sets

Challenging Problems for Chapter 4

5. Continuous Functions

Introduction to Limits

Properties of Limits

Limits Superior and Inferior

Continuity

Properties of Continuous Functions

Uniform Continuity

Extremal Properties

Darboux Property

Points of Discontinuity

Challenging Problems for Chapter 5

6. More on Continuous Functions and Sets

Introduction

Dense Sets

Nowhere Dense Sets

The Baire Category Theorem

Cantor Sets

Borel Sets

Oscillation and Continuity

Sets of Measure Zero

Challenging Problems for Chapter 6

7. Differentiation

Introduction

The Derivative

Computations of Derivatives

Continuity of the Derivative?

Local Extrema

Mean Value Theorem

Monotonicity Dini Derivatives

The Darboux Property of the Derivative

Convexity

L'Hopital's Rule

Taylor Polynomials

Challenging Problems for Chapter 7

8. The Integral

Introduction

Cauchy's First Method

Properties of the Integral

Cauchy's Second Method

Cauchy's Second Method (Continued)

The Riemann Integral

Properties of the Riemann Integral

The Improper Riemann Integral

More on the Fundamental Theorem of Calculus

Challenging Problems for Chapter 8

9. Sequences and Series of Functions

Introduction

Pointwise Limits

Uniform Limits

Uniform Convergence and Continuity

Uniform Convergence and the Integral

Uniform Convergence and Derivatives

Pompeiu's Function

Continuity and Pointwise Limits

Challenging Problems for Chapter 9

10. Power Series

Introduction Power Series: Convergence

Uniform Covergence

Functions Represented by Power Series

The Taylor Series

Products of Power Series

Composition of Power Series

Trigonometric Series

11. The Euclidean Spaces Rn

The Algebraic Structure of Rn

The Metric Structure of Rn

Elementary Topology of Rn

Sequences in Rn

Functions and Mappings

Limits of Functions from Rn to Rm

Continuity of Functions from Rn to Rm

Compact Sets in Rn

Continuous Functions on Compact Sets

Additional Remarks

12. Differentiation on Rn

Introduction Partial and Directional Derivatives

Integrals Depending on a Parameter

Differentiable Functions

Chain Rules

Implicit Function Theorems

Functions from R to Rm

Functions from Rn to Rm

13. Metric Spaces

Introduction

Metric Spaces--Specific Examples

Convergence

Sets in a Metric Space

Functions

Separable Spaces

Complete Spaces

Contraction Maps

Applications of Contraction Maps (I)

Applications of Contraction Maps (II)

Compactness

Baire Category Theorem

Applications of the Baire Category Theorem

Challenging Problems for Chapter 13

Appendix A: Background

Should I Read This Chapter?

Notation

What Is Analysis?

Why Proofs?

Indirect Proof

Contraposition

Counterexamples

Induction Quantifiers

Appendix B:

Hints for Selected Exercises

Subject Index