Real Analysis: Foundations

1. Front Matter

   Pages i-xii

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2. Rational Numbers

   • Sergei Ovchinnikov

   Pages 1-30

3. Real Numbers

   • Sergei Ovchinnikov

   Pages 31-65

4. Continuous Functions

   • Sergei Ovchinnikov

   Pages 67-86

5. Differentiation

   • Sergei Ovchinnikov

   Pages 87-106

6. Integration

   • Sergei Ovchinnikov

   Pages 107-131

7. Infinite Series

   • Sergei Ovchinnikov

   Pages 133-148

8. Back Matter

   Pages 149-178

   PDF

This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis.

Comprising six chapters, the book opens with a rigorous presentation of the theories of rational and real numbers in the framework of ordered fields. This is followed by an accessible exploration of standard topics of elementary real analysis, including continuous functions, differentiation, integration, and infinite series. Readers will find this text conveniently self-contained, with three appendices included after the main text, covering an overview of natural numbers and integers, Dedekind's construction of real numbers, historical notes, and selected topics in algebra.

Real Analysis: Foundations is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one-semester course on elementary real analysis, as well as independent study.

• Real analysis ordered fields
• Real analysis rational numbers
• Real analysis textbook
• Real analysis foundations textbook
• Real analysis logic textbook
• Real analysis intro textbook
• Real analysis real numbers
• Real analysis completeness
• Real analysis logic
• Completeness axioms
• Continuous functions
• Infinite series
• intermediate value theorem
• Convex functions real analysis
• Absolute convergence
• Riemann integral
• Darboux integral
• Dedekind's construction real numbers
• Ordered fields

“I would happily recommend to someone already familiar with the topics in RA and who is looking for an overview of the theoretical bedrock of the subject. … this text is best used by someone already quite mathematically mature, looking to further their own understanding of the Real Analysis landscape through self-study.” (John Ross, MAA Reviews, January 6, 2023)

“The text constitutes an interesting presentation … where the notion of ‘real number’ is carefully examined. … The material presented in this book will help the reader to realize why we finally decided on the theory of real functions for the standard set R of real numbers, rather than for any other ordered field. As such, it is our belief that any mathematician interested in the foundations and/or history of real analysis would benefit from going over this text.” (Krzysztof Ciesielski, Mathematical Reviews, April, 2022)

• Department of Mathematics, San Francisco State University, San Francisco, USA

  Sergei Ovchinnikov

​Sergei Ovchinnikov is Professor Emeritus of Mathematics at San Francisco State University. His other Universitext books are Functional Analysis (2018), Measure, Integral, Derivative (2013), and Graphs and Cubes (2011).

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