# The Real Numbers and Real Analysis

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 basic contents, which correspond with that introductory course, consists of seven chapters, namely the numbers 2, 3, 4, 5, 8, 9 and 10. The first chapter devoted to the real numbers, which are introduced axiomatically, and their properties. This chapter contains also a rigorous presentation of mathematical induction and recursively defined functions. The chapter concludes with a deep study of the Least Upper Bound Property and several of its consequences.

The second chapter is devoted to the limits of functions and continuity. It includes the proofs of the Extreme Value and Intermediate Value Theorems. Moreover, it is proved that these theorems are equivalent to the Least Upper Bound Property. This result is obviously very interesting, but its proof might be omitted in a basic course.

The treatment of differentiation is the standard one. The chapter includes the Rolle's Theorem and the Mean Value Theorem, as well as increasing and decreasing functions, local extrema and convex - called concave up in the text - functions.

In the chapter devoted to integration the author presents the Riemann integral, proving the Fundamental Theorem of Calculus and Lebesgue's Theorem, characterizing Riemann integrable functions as those which have a zero measure set of discontinuity points. In my opinion, introducing Riemann sums with respect to a partition and a representative set of the partition contributes nothing new with respect to the more usual way throughout upper and lower sums.

The study of sequences and series of real numbers is presented in two chapters that include the Bolzano Weierstrass Theorem and the completeness of the real numbers field. A proof of the equivalence between these results and the Least Upper Bound Theorem is given. Other results as Cantor's Lemma and the usual series convergence criteria are presented too. The author's option choice is to delay the study of sequences after the development of the course main core. It is a personal decision which has some advantages as the fact that one may immediately present some applications. But the early introduction of the sequences is, in my opinion more adequate. The ε-n arguments appearing while studying sequences, are a good introduction of the ε-δ arguments, essential for the theoretic study of continuity, differentiation and integration. On the other hand, using Bolzano Weierstrass Theorem in order to prove classical results instead of the Least Upper Bound Property has the advantage of an easier generalization of the results to higher dimensions.

This basic part ends with a chapter devoted to the study of sequences and series of functions. These concepts had been suggested before, throughout the study of power series. The chapter includes pointwise and uniform convergence, Taylor series, and ends with the appealing and interesting example of a continuous but nowhere differentiable function.

In order to provide a solid background in analysis to future secondary school teachers, the book includes the following topics. A first chapter introducing natural, integer, and rational numbers in a standard way, and the real numbers throughout Dedeking Cuts. This way is necessary once the study of sequences has been delayed. Both methods, Dedeking Cuts and Cauchy Sequences, contribute interesting ideas for the understanding of the real numbers and therefore for analysis.

I consider very interesting the chapter devoted to the trigonometric functions, to the exponential and the logarithm. The logarithm is defined by the integral, defining later the exponential as its inverse. In order to introduce the trigonometric functions, the starting point is the arcsine, which again is introduced throughout the integral. The main interest of this chapter is that it closes the gap existing between the elementary definitions and the analytic treatment of such functions.

Another good decision is the inclusion of the study of areas and lengths, which sometimes is delayed to a several variables course, or even to a Measure Theory course.

Two last remarks. The book has an interesting and useful collection of exercises; they are in general achievable for the students, but on the other hand, they also go deeply into the theory, completing the results introduced previously. Last but not least, the historic notes are excellent.

One more word, 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.

Reviewer:
Juan Ferrera
Affiliation: