Analysis with an Introduction to Proof (3rd Edition)

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9780130898791: Analysis with an Introduction to Proof (3rd Edition)

Carefully focused on reading and writing proofs, this introduction to the analysis of functions of a single real variable helps readers in the transition from computationally oriented to abstract mathematics. It features clear expositions and examples, helpful practice problems, many drawings that illustrate key ideas, and hints/answers for selected problems. Logic and Proof. Sets and Functions. The Real Numbers. Sequences. Limits and Continuity. Differentiation. Integration. Infinite Series. Sequences and Series of Functions. For anyone interested in Real Analysis or Advanced Calculus.

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A solid presentation of the analysis of functions of a real variable -- with special attention on reading and writing proofs.

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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 very difficult, 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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