METHODS OF MATHEMATICAL PROOF, PART 1 147But a case can be made that students should not view proof writing to
be so difficult as many do. For one thing, undergraduate students are, for
the most part, spared the difficulty that both haunts and delights research
mathematicians, namely, lack of certainty whether the result they are trying
to prove is actually true. Furthermore, proofs that students are asked to
write at the sophomore-senior level are usually not "hard," from the point
of view of their mathematical content. Finally, and most relevant to our
work in this chapter, there are a number of underlying principles and tech-
niques involving proof writing per se that experienced mathematicians take
for granted. Proof writing can never be reduced to a mechanical process,
but considerable anxiety and uncertainty can be eliminated from the
process-indeed, much of the "mystique" can be removed from the entire
activity of proof writing-if students are exposed to these principles and
techniques explicitly and systematically.
After some probing, students who utter the complaint at the end of the
first paragraph will often remark, "I just don't know where to begin." There
is more validity to this complaint than sometimes meets the eye, for many
elementary proofs are essentially completed once the proper starting point
has been found, once what has to be proved is carefully written down. This
step usually involves the careful interpretation of a definition. This is the
point at which problems can occur, even for students who were expert at
writing proofs in plane geometry and in deriving trigonometric identities.
A major difficulty in writing proofs at the postcalculus level is that, at this
level, the logical structure of many of the definitions encountered becomes
rather complicated. As examples, to prove that a function f is increasing
on a interval I, we must show "for pair of numbers a, b E I, a < b
implies f(a) < f(b)," a statement involving both the universal quantifier
and implication arrow. To prove that an integer m divides an integer n,
we must prove that there exists an integer p such that rt. = mp, a statement
involving the existential quantifier. To verify the definition of limit in a
particular case, we need to work with the epsilon-delta definition, involving
three uses of quantifiers (two universal and one existential), and one use of
the implication connective. Different proofs require different starting points,
a different "setting up" according to the logical structure of the conclusion
to be derived. In particular, as soon as either an existential quantifier or
an implication auow occurs in a definition involved in a desired conclusion,
the proof will involve a setting up different from anything encountered in
plane geometry or trigonometry, or in writing the few proofs that might
be required in elementary and intermediate calculus.
In this chapter we will deal systematically with various techniques of
proof, categorized largely according to different possibilities for logical struc-
ture of definitions. Since it is fruitless to discuss such techniques in a
vacuum, we will illustrate each category of proof by considering specific
mathematical problems, involving concepts that either should be familiar