6.4 PREVIEW OF ADDITIONAL ADVANCED METHODS OF PROOF (OPTIONAL) 223of the mathematician's craft. In all likelihood, if you are thoroughly ac-
quainted with, and well practiced in, the methods of proof covered thus
far, you are well along the way to a command of basic proof-writing skills.
Several other advanced methods of proof are not covered in detail in
this text. These methods, including counting arguments, compactness argu-
ments, and arguments using various transfinite processes, are general in the
sense that they have application to a wide variety of topics, yet are more
specialized and less basic than the methods of proof on which we've fo-
cused. Their applications are found primarily at the senior and graduate
levels, and their very introduction requires, for proper illustration, mathe-
matical background material that is a normal part of the junior-senior
curriculum and is, in any case, outside the domain of this text. In the re-
mainder of this article, however, we will preview these methods briefly. It
is not intended that you attempt, much less master, such proofs at this
point. Rather the purpose of this section is to heighten your awareness
that more advanced methods of proof exist and to indicate stages of future
study where such methods are likely to be encountered.
The following introduction is divided into two major categories: proofs
involving finiteness and methods based on transfinite processes.
PROOFS INVOLVING FINITENESSCounting arguments. Proofs involving counting methods are especially
prominent in relation to finite algebraic structures, particularly finite groups
and rings. You will encounter, in almost any introductory abstract algebra
course, results such as Lagrange's theorem ("the order of any subgroup of
a finite group divides the order of the group"), the theorem asserting that
"any finite integral domain is a field"; the third Sylow theorem of group
theory (whose rather technical statement we omit); and the theorem as-
serting that "the set of nonzero elements of the ring of integers modulo n
that are not divisors of zero forms a group under multiplication modulo
n," whose proofs employ counting methods. A basic counting principle
that has relevance for several of the preceding proofs is the so-called pigeon-
hole principle: If m objects are distributed among n places, where m > n,
then at least one place must receive more than one object.
Compactness arguments. Just as counting arguments occur primarily in
relation to algebra, compactness arguments occur in the domain of analysis
and topology. A student is likely to encounter the notion of compactness
for the first time in a course in advanced calculus, in the form of the Heine-
Bore1 theorem. This theorem, in its application to the real line, asserts that
any closed and bounded interval J in R has the following property, known
as compactness: "Any collection of open intervals that cover J has a finite
subcollection that also covers J." This admittedly technical property is the
theoretical basis for the proofs of a pair of theorems whose statements may
already be familiar to you: "A continuous function on a closed and bounded