Methods of
Mathematical Proof,
Part 11.
~dvanc& Methods
CHAPTER 6
The question of what constitutes an advanced, as opposed to elementary,
method of mathematical proof is largely subjective. In truth, the difficulty
of a given proof depends more on the details of the particular theorem than
on the logical structure of the conclusion of that theorem. Our criteria for
inclusion under the "advanced" designation are based partly on experience
with students' reaction to various kinds of proofs, and partly on the difficulty
of the applications involved in illustrating these categories. On this basis,
proofs involving existence, uniqueness, and various indirect methods are
categorized in this text as advanced. Many of the exercises in this chapter
are more difficult, and specialized, than those in Chapter 5; some may be
appropriate primarily for those who have already had experience in junior-
senior level courses.
6.1 Conclusions Involving V, Followed by 3
(Epsilon-Delta Proofs Optional)Many important definitions in mathematics involve the existential quantifier
- Such definitions are virtually nonexistent in precalculus mathematics
and occur relatively infrequently in the standard calculus sequence. The
best known of these is the epsilon-delta definition of limit, discussed earlier