(b) Let f be a function that maps real numbers to real numbers, and let A and B
be subsets of the domain off. Prove that f (A n B) G f (A) n f (B): Give an ex-
ample to show that the reverse inclusion need not hold.
(c) Under the assumptions of (b), prove that f (A u B) = f (A) u f (B).
(d) Under the assumptions of (b), prove that if A z B, then f(A) z f(B).- In Example 10 the well-ordering principle for the set N of all positive integers
was introduced and used to prove the existence of a greatest common divisor (m, n)
for any integers m and n, not both zero.
(a) Prove that if m and n are integers, not both zero, then they have a unique
greatest common divisor.
(b) Use the proof in Example 10 to show that if d = (m, n), then there exist integers
x and y such that d = mx + ny.
(c) Use the result of (b) to show that if a, b, and c are integers such that a 1 bc and
(a, b) = 1, then a lc (recall Exercise 2, Article 6.2 and Exercise 2, Article 6.1).
(d) Use the well-ordering principle to prove that if S is a subset of the set N of
all positive integers satisfying these two properties: (i) 1 E S and (ii) for all m E N,
if m E S, then m + 1 E S; then S = N. This is known as the principle of mathe-
matical induction (recall Article 5.4). (Hint: if S # N, then S is a proper subset
of N so that N - S is a nonempty subset of N.)
10. One of the basic properties of the real number system is the least upper bound
axiom: Every nonempty subset of R that is bounded above in R has a least upper
bound in R (cf., Examples 4 and 7). We will see in Article 9.3 that this axiom is
one of the basic distinguishing features between the real and rational number sys-
tems; that is, Q fails to satisfy this axiom.
"(a) Use the least upper bound axiom to derive the Archimedean property: If
a and b are any positive real numbers, there exists a positive integer n such
that b < na. (Hint: Suppose the conclusion is false and consider the set S =
{nu ( n E N) .)
(b) Use the result in (a) to prove that the sequence {l/n) converges to zero. (Note:
This result has previously been assumed in exercises such as Exercise 20, Article
6.1.)
(c) Use the result in (a) to prove that the set N of all positive integers is not
bounded above in R.
(d) Use the least upper bound axiom and ideas suggested by Exercise 7(b) to prove
that a nonempty set of real numbers that is bounded below in R has a greatest
lower bound in R. (First formulate, on the basis of the definition of "least upper
bound," in Example 4, an appropriate definition of "greatest lower bound.")
*(e) Recall from Exercise 9, Article 6.1, the definition of "point of accumulation of
a set S." Show that if u = lub S and u $ S, then u is a point of accumulation of
S. Give an example to show that lub S need not in general be a point of accumu-
lation of S.
- A basic theorem of mathematical analysis asserts that iff is a function that is
continuous on a closed and bounded interval [a, b], then f attains both an absolute
maximum and minimum value on that interval. The first of these means that there
exists m E [a, b] such that f(m) 2 f(x) for all x E [a, b]. You should formulate the
second definition.