Calculus: Analytic Geometry and Calculus, with Vectors

5.6 Sequences, series, and decimals 343

Show that, for each n = 1, 2, 3, ,

(2) an < an+1 < bn+1 < b,.

Tell why there must exist numbers L1 and L2 such that

(3)

Show also that

(4)

and hence

(5)

lim an = L1, lim b, = L2.

n---). 8-100

an+bn bn - an

0 <bn+1-an+1 < 2 -an= 2

0 < L2 - L1 < (L2 - L1),

so L2 = L1. Remark: The common value of the two limits in (3) has an impressive

name; it is the arithmetico-geometric mean of the two given numbers a1 and b1.

11 This problem, which is in some respects the most significant problem in

this chapter, would be much too difficult if it were not prefaced by a rather elab-

orate story. We make the reasonable assumption that Mr. C., a particular

carpenter, never heard of the Dedekind axiom 5.43, and that his ideas about the

real-number system are incomplete. Next we make the reasonable assumption

that the class R* (read R star) of numbers that Mr. C. knows about is the class

of rational numbers which he may call "whole numbers and fractions." This

class R* is, for many purposes, a thoroughly useful class of numbers. If x and

y belong to R*, so do x + y, x - y, xy, and also x/y, provided y 0 0. While

we may be somewhat surprised by the fact, it is nevertheless true that Mr. C.

could define graphs, limits, derivatives, indefinite integrals, Riemann integrals,

and many other things exactly as we defined them. Mr. C. could show, exactly

as we did, that if f(x) = x2, then f'(x) = 2x. There would be many respects in

which his analytic geometry and calculus would be thoroughly satisfactory.

He would say, exactly as we did, that f is continuous at xo if to each e > 0 there

corresponds a S > 0 such that lf(x) - f(xo)I < e whenever Ix - xol < 5, but of

course only rational numbers appear in his work. Mr. C. would be totally

unaware of the existence of irrational numbers, but we could nevertheless select

an irrational number for which 0 < < 1 and put Mr. C. to work studying

the function f for which

(1) J f(x) = - 1 (0 5 x < )

lf(x)=1 ( <x=<1).

Mr. C. would discover that f is defined for each x in R for which 0 < x < 1,
and hence he would say that it is defined over the interval 0 < x 5 1. He could
prove that f is continuous at each x in R for which 0 < x < 1. He would
therefore say that it is continuous over the interval 0 < x 5 1. He could prove
that f(x) = 0 for each x in R* for which 0 < x < 1. So far there is nothing
wrong, but there will be something wrong if Mr. C. tries to tell us that f'(x) = 0
when 0 < x < 1 and hence "it is obvious" or "it can be shown" that there must
be a constant k such that f(x) = k when 0 < x < 1. In fact, a look at the for-
mulas (1) defining f shows that there is no constant k such that f(x) = k when