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Über dieses Buch

Lively prose and imaginative exercises draw the reader into this unique introductory real analysis textbook. Motivating the fundamental ideas and theorems that underpin real analysis with historical remarks and well-chosen quotes, the author shares his enthusiasm for the subject throughout. A student reading this book is invited not only to acquire proficiency in the fundamentals of analysis, but to develop an appreciation for abstraction and the language of its expression.
In studying this book, students will encounter:the interconnections between set theory and mathematical statements and proofs;
the fundamental axioms of the natural, integer, and real numbers;
rigorous ε-N and ε-δ definitions;
convergence and properties of an infinite series, product, or continued fraction;
series, product, and continued fraction formulæ for the various elementary functions and constants.
Instructors will appreciate this engaging perspective, showcasing the beauty of these fundamental results.



Some Standard Curriculum


Chapter 1. Very Naive Set Theory, Functions, and Proofs

One of the goals of this text is to get you proving mathematical statements in real analysis. Set theory provides a safe environment in which to learn about math statements, “if ... then,” “if and only if,” etc., and to learn the logic behind proofs. Since this is an introductory book on analysis, our treatment of sets is “very naive,” in the sense that we actually don’t define sets rigorously, only informally; we are mostly interested in how “they work,” not really what they are.
Paul Loya

Chapter 2. Numbers, Numbers, and More Numbers

This chapter is on the study of numbers. Of course, we all have a working understanding of the real numbers, and we use many aspects of these numbers in everyday life: tallying up tuition and fees, figuring out how much we have left on our food cards, etc. We have accepted from our childhood all the properties of numbers that we use every day.
Paul Loya

Chapter 3. Infinite Sequences of Real and Complex Numbers

Notable enough, however, are the controversies over the series \(1 - 1 + 1 - 1 + 1 -\).
Paul Loya

Chapter 4. Limits, Continuity, and Elementary Functions

In this chapter we study what are without doubt the most important functions in all of analysis and topology, the continuous functions.
Paul Loya

Chapter 5. Some of the Most Beautiful Formulas in the World I–III

In this chapter we present a small sample of some of the most beautiful formulas in the world.
Paul Loya

Extracurricular Activities


Chapter 6. Advanced Theory of Infinite Series

This chapter is about going in depth into the theory and application of infinite series. One infinite series that will come up again and again in this chapter and the next chapter as well is the Riemann zeta function.
Paul Loya

Chapter 7. More on the Infinite: Products and Partial Fractions

This chapter is devoted entirely to the theory and application of infinite products, and as a consolation prize, we also talk about partial fractions.
Paul Loya

Chapter 8. Infinite Continued Fractions

We dabbled a little into the theory of continued fractions (that is, fractions that continue on and on and on ...) way back on the exercises of Section 3.​4. In this chapter we concentrate on this fascinating subject.
Paul Loya


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