What happens when a student completes a course in “AP Calculus”, followed
by a routine course in multivariable calculus, a computational course in
linear algebra, and a formulaic presentation of differential equations? It is
time for some real mathematics. There is still a world of interconnected
labyrinths to explore, unscalable mountains to be climbed, and lands of
mystery to be discovered. The first step in this is a rigorous course in one-
variable analysis. This begins with a study of an ordered field in which
the least upper bound property holds (not to be confused with a complete
ordered field; see the project in Section 2.7.3). Here the student meets
Bolzano-Weierstrass, Heine-Borel, and a rigorous treatment of one-variable
differentiation and integration with careful attention paid to the pervasive
presence of the Mean Value Theorem. Then what? That is exactly the
reason behind this book.
Learning serious mathematics is about engaging with problems, from
kindergarten to graduate school and beyond. The preliminaries for reading
this book are already contained in the author’s book Tools of the Trade .
The first two chapters of Tools are Appendices A and B of this book. The
reader who is familiar with that material can jump right into Chapter 1.
The sequence of topics can be gleaned from the table of contents, so I will
not dwell on that.
There are three features here that should be discussed explicitly, espe-
cially since they are important for the use of this book as a text. The first,
and most important, is the collection of exercises. These are spread through-
out the chapters and should be regarded as an essential component of the
student’s learning. Some of these exercises comprise a routine follow-up to
the material, while others will challenge the student’s understanding more
deeply. The second feature is the set of independent projects presented at
the end of each chapter. These projects supplement the content studied in
their respective chapters. They can be used to expand the student’s knowl-
edge and understanding or as an opportunity to conduct a seminar in Inquiry
Based Learning (IBL) in which the students present the material to their
class. A brief glance will show that the independent projects cover a wide
range of interesting topics that hint at advanced areas of mathematics. The