The Real Numbers and Real Analysis, 2011

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.

Provides an unusually thorough treatment of the real numbers, emphasizing their importance as the basis of real analysis

Presents material in an order resembling that of standard calculus courses, for the sake of student familiarity, and for helping future teachers use real analysis to better understand calculus

Emphasizes the direct role of the Least Upper Bound Property in the study of limits, derivatives and integrals, rather than making use of sequences for proofs

Presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property

Relates real analysis to previously learned materal, including detailed discussion of such topics as the transcendental functions, area and the number pi

Offers three different entryways into the study of real numbers, depending on the student audience

Contains historical context, biographical anecdotes, and reflections on the material in each chapter

Includes over 350 exercises, reinforcing concepts

Includes supplementary material: sn.pub/extras