Introduction When structuring an undergraduate mathematics program, ordinarily the faculty designs the initial set of courses to provide techniques that permit a student to solve problems of a more or less computational nature. So, for example, students might begin with a one variable calculus course and proceed through multi-variable calculus, ordinary differential equations, and linear algebra without ever encountering the fundamental ideas that underlie this mathematics. If the students are to learn to do mathematics well, they must at some stage come to grips with the idea of proof in a serious way. In this book, we attempt to provide enough background so that students can gain familiarity and facility with the mathematics required to pursue de- manding upper-level courses. The material is designed to provide the depth and rigor necessary for a serious study of advanced topics in mathematics, especially analysis. There are several unusual features in this book. First, the exercises, of which there are many, are spread throughout the body of the text. They do not occur at the ends of the chapters. Instead Chapters 1–4 close with special projects that allow the teachers and students to extend the material covered in the text to a much wider range of topics. These projects are an integral part of the book, and the results in them are often cited in later chapters. They can be used as a regular part of the class, a source of independent study for the students, or as an Inquiry Based Learning (IBL) experience in which the students study the material and present it to the class. At the end of Chapter 5, there is a collection of Challenge Problems that are intended to test the students’ understanding of the material in all five chapters as well as their mathematical creativity. Some of these problems are rather simple while others should challenge even the most able students. ix
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