xxii To the Student
You will recall that calculus is built upon the concept of limit. Thus, Chap-
ter 2 takes up that idea. Along with many other analysts, I believe that limits
are best understood and appreciated if they are introduced in the context of
infinite sequences. You will probably be quite amazed at the power of this ap-
proach to limits. In fact, you might come to agree with me that Chapter 2 is
the key chapter in the whole course. Chapters 2 and 3 contain many important
ideas upon which the whole subject of analysis depends.
In Chapter 4 we take up limits of functions, and in Chapter 5 we define and
derive deep results about the related notion of continuous functions. The notion
of continuity is subtle yet far more powerful than you might expect. Chapter 6 at
last looks like "calculus;" in it the familiar rules and properties of differentiable
functions are derived using the techniques of limits. In particular, we see why
the mean value theorem is so important, and we see Taylor's Theorem in its
proper context as a type of mean value theorem. All along, we continue to rely
on the techniques developed in Chapter 2.
Chapter 7 is a fresh start. In it we develop the Riemann integral from
scratch, using only the properties of the real number system developed in Chap-
ter 1 until we come to the point where we wish to relate it to the previously
learned concepts of continuity and the derivative. The fundamental theorem
of calculus is a natural high point, tying together the major themes of the
course, but once again supported by the basic concepts of limits established in
Chapter 2.
In Chapter 8 you will renew your acquaintance with infinite series, a topic
you previously studied in calculus. This time you will examine series in greater
depth and hopefully will find them more interesting and useful than you had
previously believed. We will uncover some quite interesting facts and techniques
involving series.
In Chapter 9 functions are considered as objects themselves, or points in a
"function space." Considering sequences and series of functions (rather than of
numbers) leads to many powerful results in higher level analysis. We conclude
this chapter with two especially intriguing results: the existence of continuous,
nowhere differentiable functions, and the fact that every continuous function
can be approximated with any prescribed accuracy by a polynomial.
REVIEW MATERIAL IN APPENDICESThe statements and proofs of analysis often involve a high degree of complexity
and subtlety. You will find these statements and proofs much less intimidating
if you have a working knowledge of the principles of formal logic. Familiarity
with the rules of logic will minimize confusion and improve the clarity of the
statements of analysis. Many students acquire this familiarity in a "transitional"
or proof-oriented mathematics course.
Some of the actual symbolism of "symbolic logic" is used in this book.
The purpose of this symbolism is not to confuse, but to clarify mathematical