xxvi To the Instructor
In her essay [56] quoted at the beginning of this section, Grabiner points
to another factor contributing to the increased attention paid to rigor in the
works of Cauchy, Riemann, Weierstrass, and their contemporaries. That was
"the need to teach." As the number of royal courts with sufficient affluence
to employ resident mathematicians declined, the number of mathematicians
was on the increase. Forced to look elsewhere for employment, mathematicians
found that they were needed to teach in technical high schools and universities.
In Grabiner's words, "Teaching forces one's attention to basic questions,'' and
"provide[s] a catalyst for the crystallization of the foundations of the calcu-
lus .... "
In our own times, real analysis is being taught to an ever-widening audience.
Since calculus is universally taught in American high schools, it is now crucially
important that teachers certified to teach secondary school mathematics have a
firm understanding of at least the elements of real analysis. Thus many colleges
and universities (such as my own) are requiring all their mathematics majors
to take a course in elementary real analysis. This book is written to fulfill the
need for a textbook appropriate for such a course.
Teaching elementary real analysis can be quite challenging. Typical under-
graduates find the concepts and techniques of analysis difficult. The pleasure
you find in the subject, and the enthusiasm with which you teach it, may be
met by somewhat disappointing results. This book was written out of my de-
sire to improve those results by making the material more understandable to
the average student. I am assuming that your typical student is not too much
different from mine.
As teachers we need to recognize that in order for students to grow they
must start from where they are. Mathematical maturity will come, but rarely
does it appear full-blown from the start. This book takes a more patient ap-
proach. I ask for your patience too; I believe you will like the results.
STYLE OF PROOFSBeauty is in the eye of the beholder! That includes mathematical beauty. What
the professional mathematician finds so satisfying in an elegant proof may be
inaccessible to a novice undergraduate. A student may be left completely in the
dark by a proof presented in the sli ck style of a professional mathematician.
Students need to see proofs presented in a style that they can understand and
emulate in their own proofs. In this way they are much more likely to become
comfortable reading and constructing proofs.
Because proofs in this book are written for digestion by students, you might
initially find them somewhat tedious and over-detailed. They are designed to
facilitate gradual but steady growth, which is a more reasonable expectation
than instantaneous maturation. Features of this style include:- The hypotheses of a theorem are stated at the beginning of each proof.
This may seem redundant, as they are already stated in the theorem itself.