6.1 CONCLUSIONS INVOLVING V, FOLLOWED BY 3 199it works! The best answer we can give to the second question is that most
students have considerable difficulty in writing epsilon-delta proofs (largely
because of deficiencies in logical background and lack of experience in writ-
ing any kind of proof-the very weaknesses you may be eliminating as
you work through this text) and only begin to overcome that difficulty with
much effort and experience. Although all the information needed to write
proofs such as those in Exercises 12 through 19 has been provided in this
article, do not be surprised or discouraged if you have difficulty in your
early attempts. Finally, even after you have mastered these proofs, through
a process of studying a number of similar proofs written by others and
your own maturing in mathematics, keep in mind that a still higher level
to attain is the ability to discover, on your own, original proof techniques
(such as the use of ~/2) and perhaps, at some stage, prove theorems pre-
viously unproved and solve mathematical problems previously unsolved.
This pursuit is the goal of mathematical study at the graduate level; an
undergraduate student who is able to do many of the following exercises,
shows promise of possible success at that level.
Exercises
- Use the definition in Example 1 and the ordinary rules for addition and multi-
plication of fractions, together with the fact that Z is closed under addition and
multiplication, to prove
(a) If x and y are rational, then xy is rational
(6) If x and y are rational, then x + y is rational
(c) If x and y are rational and y # 0, then x/y is rational- Use the definition in Example 2 to prove that, for any m, n, p E Z
(a) Ifmln and nip, then m(p
(6) Ifmlnandnlm, then n=mor n= -m
(c) mlm (d) 1Jn
(4 m10 (f) Ifmlnandmlp,thenmlnp
(g) 1f mln, then ml(-n)
.
(h) Based on your proof in (f), can you improve upon the result in (f), that is,
state and prove a stronger theorem (due to a weaker hypothesis) than the one
in (f)? (Note: Reread the fourth and fifth paragraphs under the heading "Math-
ematical Significance of Tautologies Involving the Conditional," following
Theorem 2, Article 2.3. Recall also Exercise 7, Article 2.3.)- (a) Prove that if U = Z, then: