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This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus.
The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.
List of contents
Preface.-To the Student.-To the Instructor.- 1. Construction of the Real Numbers.- 2. Properties of the Real Numbers.- 3. Limits and Continuity.- 4. Differentiation.- 5. Integration.- 6. Limits to Infinity.-7. Transcental Functions.-8. Sequences.- 9. Series.- 10. Sequences and Series of Functions.- Bibliography.- Index.
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From the reviews:
"The author's purpose is to cover with this book the necessary mathematical background for secondary school teachers. The book is also useful for an introductory one real variable analysis course. ... The book has an interesting and useful collection of exercises ... . Last but not least, the historic notes are excellent. ... I consider this book of great interest for the academic training of the future secondary school teachers, so the author's purpose is greatly fulfilled." (Juan Ferrera, The European Mathematical Society, April, 2013)
"Bloch (Bard College) has written an introductory book on analysis at the undergraduate level, with enough material for at least two semesters of studies. The author writes very carefully and includes numerous examples and historical insights. The exposition is generally excellent. The book provides all proofs with enough details for most undergraduates to follow through without undue difficulties... Overall, an excellent book. Summing Up: Highly recommended. Upper-division undergraduates, graduate students, and faculty."
-D. M. Ha, Ryerson University, Choice, February 2012
"The most distinctive characteristic of this text on real analysis is its three-in-one feature. It was designed specifically for three distinct groups of students. ... The book was motivated by a need for a textbook for the M.A.T. students, but is intended to have enough flexibility to serve the other groups as well. ... this is a strong text, especially for students who need more guidance and support. The book gives an instructor plenty of options for planning a course." (William J. Satzer, The Mathematical Association of America, August, 2011)
