To the Student xxitake a critical, even skeptical approach. Mere passive acceptance will not do! In
fact, we will be so critical that we will not consider any statement of analysis to
be fully reliable until we have a firm justification of it (which we call a "proof").
For this justification, we are forced to look backward to the foundations of the
subject. In one sense, you are asked to forget what you learned about calculus
(if you haven't already) and build upon a new foundation.WHY PROOFS ARE IMPORTANT"Proof' can be an intimidating word to many students of mathematics who
would rather just be told what is true. The reason why proof is so important
in mathematics is found in the very nature of "mathematical truth" itself, as
understood in the western intellectual tradition. In this tradition, a body of
mathematics is not just a collection of disconnected "facts" that are accepted
because they seem to be true. Rather, these facts must be connected together
and organized according to the "deductive method." Mathematicians go to
great lengths to isolate some of the facts that they can regard as basic (com-
ing at the very beginning) and then go to even greater lengths to show that
all the remaining facts can be derived from the basic ones by the process of
logical deduction. The "facts" that come at the beginning are then really as-
sumptions (axioms). The remaining facts, as they are deduced one-by-one from
these assumptions, are called "theorems." The process of deducing (or deriving)
a theorem is called "proof."
What, then, does a proof prove? The answer is not as obvious as it may
seem at first. A theorem is ultimately derived from the axioms set forth at the
beginning of a mathematical subject. Thus, the truth of a theorem is really
contingent upon the truth of the axioms. If the axioms are all true, then any
theorem that is derived from them by valid logical deduction must also be
true. But the proof of a theorem cannot assure us that the axioms upon which
the proof rests are themselves true. Thus, a "proof" does not guarantee that a
theorem is true. A proof guarantees only that if the axioms are all true, then the
theorem is true. In other words, a proof of a theorem proves that the axioms
are sufficiently strong to guarantee the theorem. Thus, a theorem is really a
statement about the axioms. For this reason axioms serve as the foundation of
a mathematical subject.
OUR PLAN OF ATTACKIn Chapter 1 we set forth a few basic assumptions from which the entire subject
of analysis can be derived by the process of logical deduction. In later chapters
we carry out that process as far as the constraints of time permit. We reach
a natural culmination point with the "fundamental theorem of calculus," in
Chapter 7. Readers who continue to the end of the book will reach another
culmination point in Chapter 9, exploring some fascinating consequences of
uniform convergence.