326 CHAPTER 9 CALCULUS
9.2. 13 Let (an) be a (possibly infinite) sequence of
positive integers. A creature likeao + -----;---
a( + I
a 2 +--I
a 3 +-is called a continued fraction and is sometimes de
noted by [ao,a(,a 2 ," ']'
(a) Give a rigorous interpretation of the number
[ao,a( ,a 2 ,·· .].
(b) Evaluate [1,1,1,. .. ].
(c) Evaluate [1,2, 1,2, ... ].
(d) Find a sequence (an) of positive integers such
that v'2 = [ao,a(,a 2 , ... ]. Can there be more
than one sequence?
(e) Show that there does not exist a repeating se
quence (an) such that
V'2 = [ao,a( ,a 2 ," .].9.2. 14 Carefully prove the assertion stated on
page 318, that all bounded monotonic sequences con
verge.
9.2. 15 Dense Sets. A subset S of the real numbers is
called dense if, given any real number x, there are ele
ments of S that are arbitrarily close to x. For example,
Q is dense, since any real number can be approximated
arbitrarily well with fractions (look at decimal approx
imations). Here is a formal definition:
A set of real numbers S is dense if, given
any x E lR and e > 0, there exists s E S
such that Is - xl < e.
More generally, we say that the set S is dense in the
set T if any t E T can be approximated arbitrarily well
by elements of S. For example, the set of positive frac
tions with denominator greater than numerator is dense
in the unit interval [0, I].
(a) Observe that "S is dense in T" is equivalent
to saying that for each t E T, there is an in
finite sequence (skl of elements in S such that
k lim Sk = t.
.... oo
(b) Show that the set of real numbers that are zeros
of quadratic equations with integer coefficients
is dense.(c) Let D be the set of dyadic rationals, the ratio
nal numbers whose denominators are powers of
two (some examples are 1/2,5/8, 1037/256).
Show that D is dense.
(d) Let S be the set of real numbers in [0, I] whose
decimal representation contains no 3s. Show
that S is not dense in [0, I].
9.2. 16 Define (x) := x-lxJ. In other words, (x) is the"fractional part" of x; for example (n) = (^0). (^14159) ....
Let
a = 0.1234567891011 12131415 ... ;
in other words, ao is the number formed by writing
every positive integer in order after the decimal point.
Show that the set
{(a), (lOa), (l0^2 a), ... }
is dense in [0, 1] (see Problem 9.2.15 for the definition
of "dense").
9.2. 17 Let a be irrational.
(a) Show that the sequence
(a), (2a), (3a), ...
contains a subsequence that converges to zero.
Hint: First use the pigeonhole principle to show
that one can get two points in this sequence ar
bitrarily close to one another.
(b) Show that the set
{(a), (2a), (3a), ... }
is dense in [0, I]. (See Problem 9.2.15 for the
definition of "dense" and Problem 9.2.16 for the
definition of (x).)
9.2. 18 Consider a circle with radius 3 and center at
the origin. Points A and B have coordinates (2,0) and
(-2,0), respectively. If a dart is thrown at the circle,
then assuming a uniform distribution, it is clear that
the probabilities of the following two events are equal.
- The dart is closer to A than to B.
- The dart is closer to B than to A.
We call the two points "fairly placed." Is is possible to
have a third point C on this circle, so that A, B, Care
all fairly placed?
9.2. 19 Define the sequence (an) by ao = a and
an+ ( = an - a� for n � I. Discuss the convergence
of this sequence (it will depend on the initial value a).