ISBN-10:
032174747X
ISBN-13:
9780321747471
Pub. Date:
01/05/2013
Publisher:
Pearson
Analysis with an Introduction to Proof / Edition 5

Analysis with an Introduction to Proof / Edition 5

by Steven R. Lay

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Overview

Analysis with an Introduction to Proof / Edition 5

For courses in undergraduate Analysis and Transition to Advanced Mathematics.

Analysis with an Introduction to Proof, Fifth Edition helps fill in the groundwork students need to succeed in real analysis—often considered the most difficult course in the undergraduate curriculum. By introducing logic and emphasizing the structure and nature of the arguments used, this text helps students move carefully from computationally oriented courses to abstract mathematics with its emphasis on proofs. Clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers make this text readable, student-oriented, and teacher- friendly.

Product Details

ISBN-13: 9780321747471
Publisher: Pearson
Publication date: 01/05/2013
Series: Featured Titles for Real Analysis Series
Edition description: New Edition
Pages: 400
Sales rank: 441,519
Product dimensions: 7.50(w) x 9.20(h) x 0.80(d)

About the Author

Steven Lay is a Professor of Mathematics at Lee University in Cleveland, TN. He received M.A. and Ph.D. degrees in mathematics from the University of California at Los Angeles. He has authored three books for college students, from a senior level text on Convex Sets to an Elementary Algebra text for underprepared students. The latter book introduced a number of new approaches to preparing students for algebra and led to a series of books for middle school math. Professor Lay has a passion for teaching, and the desire to communicate mathematical ideas more clearly has been the driving force behind his writing. He comes from a family of mathematicians, with his father Clark Lay having been a member of the School Mathematics Study Group in the 1960s and his brother David Lay authoring a popular text on Linear Algebra. He is a member of the American Mathematical Society, the Mathematical Association of America, and the Association of Christians in the Mathematical Sciences.

Read an Excerpt

PREFACE:

Preface

A student's first encounter with analysis has been widely regarded as the most difficult course in the undergraduate mathematics curriculum. This is due not so much to the complexity of the topics as to what the student is asked to do with them. After years of emphasizing computation (with only a brief diversion in high school geometry), the student is now expected to be able to read, understand, and actually construct mathematical proofs. Unfortunately, often very little groundwork has been laid to explain the nature and techniques of proof.

This text seeks to aid students in their transition to abstract mathematics in two ways: by providing an introductory discussion of logic, and by giving attention throughout the text to the structure and nature of the arguments being used. The first two editions have been praised for their readability and their student-oriented approach. This revision builds on those strengths. Small changes have been made in many sections to clarify the exposition, and several new examples and illustrations have been added.

The major change in this edition is the addition of more than 250 true/false questions that relate directly to the reading. These questions have been carefully worded to anticipate common student errors. They encourage the students to read the text carefully and think critically about what they have read. Often the justification for an answer of "false" will be an example that the students can add to their growing collection of counterexamples.

As in earlier editions, the text also includes more than a hundred practice problems. Generally, these problems are not verydifficult, and it is intended that students should stop to work them as they read. The answers are given at the end of each section just prior to the exercises. The students should also be encouraged to read (if not attempt) most of the exercises. They are viewed as an integral part of the text and vary in difficulty from the routine to the challenging. Those exercises that are used in a later section are marked with an asterisk. Hints for many of the exercises are included at the back of the book. These hints should be used only after a serious attempt to solve an exercise has proved futile.

The overall organization of the book remains the same as in the earlier editions. The first chapter takes a careful (albeit nontechnical) look at the laws of logic and then examines how these laws are used in the structuring of mathematical arguments. The second chapter discusses the two main foundations of analysis: sets and functions. This provides an elementary setting in which to practice the techniques encountered in the previous chapter.

Chapter 3 develops the properties of the real numbers R as a complete ordered field and introduces the topological concepts of neighborhoods, open sets, closed sets, and compact sets. The remaining chapters cover the topics usually included in an analysis of functions of a real variable: sequences, continuity, differentiation, integration, and series.

The text has been written in a way designed to provide flexibility in the pacing of topics. If only one term is available, the first chapter can be assigned as outside reading. Chapter 2 and the first half of Chapter 3 can be covered quickly, again with much of the reading being left to the student. By so doing, the remainder of the book can be covered adequately in a single semester. Alternatively, depending on the students' background and interests, one can concentrate on developing the first five chapters in some detail. By placing a greater emphasis on the early material, the text can be used in a "transitional" course whose main goal is to teach mathematical reasoning and to illustrate its use in developing an abstract structure. It is also possible to skip derivatives and integrals and go directly to series, since the only results needed from these two chapters will be familiar to the student from beginning calculus.

A thorough treatment of the whole book would require two semesters. At this slower pacing the book provides a unified approach to a course in foundations followed by a course in analysis. Students going into secondary education will profit greatly from the first course, and those going on to graduate school in either pure or applied mathematics will want to take both semesters.

I appreciate the helpful comments that I have received from users of the first two editions and reviewers of the third. In particular, I would like to thank Professors Michael Dutko, Ana Mantilla, Marcus Marsh, Carl Maxson, Stanley Page, Doraiswamy Ramachandran, Ernie Solheid, David Trautman, Kevin Yeomans, and Zbigniew Zielezny. I am also grateful to my students at Lee University for their numerous suggestions.

Steven R. Lay
Cleveland, TN

Table of Contents

1. Logic and Proof

Section 1. Logical Connectives

Section 2. Quantifiers

Section 3. Techniques of Proof: I

Section 4. Techniques of Proof: II

2. Sets and Functions

Section 5. Basic Set Operations

Section 6. Relations

Section 7. Functions

Section 8. Cardinality

Section 9. Axioms for Set Theory(Optional)

3. The Real Numbers

Section 10. Natural Numbers and Induction

Section 11. Ordered Fields

Section 12. The Completeness Axiom

Section 13. Topology of the Reals

Section 14. Compact Sets

Section 15. Metric Spaces (Optional)

4. Sequences

Section 16. Convergence

Section 17. Limit Theorems

Section 18. Monotone Sequences and Cauchy Sequences

Section 19. Subsequences

5. Limits and Continuity

Section 20. Limits of Functions

Section 21. Continuous Functions

Section 22. Properties of Continuous Functions

Section 23. Uniform Continuity

Section 24. Continuity in Metric Space (Optional)

6. Differentiation

Section 25. The Derivative

Section 26. The Mean Value Theorem

Section 27. L'Hospital's Rule

Section 28. Taylor's Theorem

7. Integration

Section 29. The Riemann Integral

Section 30. Properties of the Riemann Integral

Section 31. The Fundamental Theorem of Calculus

8. Infinite Series

Section 32. Convergence of Infinite Series

Section 33. Convergence Tests

Section 34. Power Series

9. Sequences and Series of Functions

Section 35. Pointwise and uniform Convergence

Section 36. Application of Uniform Convergence

Section 37. Uniform Convergence of Power Series

Glossary of Key Terms

References

Hints for Selected Exercises

Index

Preface

PREFACE:

Preface

A student's first encounter with analysis has been widely regarded as the most difficult course in the undergraduate mathematics curriculum. This is due not so much to the complexity of the topics as to what the student is asked to do with them. After years of emphasizing computation (with only a brief diversion in high school geometry), the student is now expected to be able to read, understand, and actually construct mathematical proofs. Unfortunately, often very little groundwork has been laid to explain the nature and techniques of proof.

This text seeks to aid students in their transition to abstract mathematics in two ways: by providing an introductory discussion of logic, and by giving attention throughout the text to the structure and nature of the arguments being used. The first two editions have been praised for their readability and their student-oriented approach. This revision builds on those strengths. Small changes have been made in many sections to clarify the exposition, and several new examples and illustrations have been added.

The major change in this edition is the addition of more than 250 true/false questions that relate directly to the reading. These questions have been carefully worded to anticipate common student errors. They encourage the students to read the text carefully and think critically about what they have read. Often the justification for an answer of "false" will be an example that the students can add to their growing collection of counterexamples.

As in earlier editions, the text also includes more than a hundred practice problems. Generally, these problems are notverydifficult, and it is intended that students should stop to work them as they read. The answers are given at the end of each section just prior to the exercises. The students should also be encouraged to read (if not attempt) most of the exercises. They are viewed as an integral part of the text and vary in difficulty from the routine to the challenging. Those exercises that are used in a later section are marked with an asterisk. Hints for many of the exercises are included at the back of the book. These hints should be used only after a serious attempt to solve an exercise has proved futile.

The overall organization of the book remains the same as in the earlier editions. The first chapter takes a careful (albeit nontechnical) look at the laws of logic and then examines how these laws are used in the structuring of mathematical arguments. The second chapter discusses the two main foundations of analysis: sets and functions. This provides an elementary setting in which to practice the techniques encountered in the previous chapter.

Chapter 3 develops the properties of the real numbers R as a complete ordered field and introduces the topological concepts of neighborhoods, open sets, closed sets, and compact sets. The remaining chapters cover the topics usually included in an analysis of functions of a real variable: sequences, continuity, differentiation, integration, and series.

The text has been written in a way designed to provide flexibility in the pacing of topics. If only one term is available, the first chapter can be assigned as outside reading. Chapter 2 and the first half of Chapter 3 can be covered quickly, again with much of the reading being left to the student. By so doing, the remainder of the book can be covered adequately in a single semester. Alternatively, depending on the students' background and interests, one can concentrate on developing the first five chapters in some detail. By placing a greater emphasis on the early material, the text can be used in a "transitional" course whose main goal is to teach mathematical reasoning and to illustrate its use in developing an abstract structure. It is also possible to skip derivatives and integrals and go directly to series, since the only results needed from these two chapters will be familiar to the student from beginning calculus.

A thorough treatment of the whole book would require two semesters. At this slower pacing the book provides a unified approach to a course in foundations followed by a course in analysis. Students going into secondary education will profit greatly from the first course, and those going on to graduate school in either pure or applied mathematics will want to take both semesters.

I appreciate the helpful comments that I have received from users of the first two editions and reviewers of the third. In particular, I would like to thank Professors Michael Dutko, Ana Mantilla, Marcus Marsh, Carl Maxson, Stanley Page, Doraiswamy Ramachandran, Ernie Solheid, David Trautman, Kevin Yeomans, and Zbigniew Zielezny. I am also grateful to my students at Lee University for their numerous suggestions.

Steven R. Lay
Cleveland, TN

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