Bridge to Abstract Mathematics: Mathematical Proof and Structures

354 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10

Proof Let E be an arbitrary positive rational number. We must prove that

there exists a positive integer N such that, for any m, n E N with m, n 2

N, we have Iq, - qnl < E. According to the hypothesis of convergence,

say to a rational number q, we may assert that, corresponding to the

given E, divided by 2, there is a positive integer No such that, for any

integers m and n greater than or equal to No, we have both Iq, - ql <

€12 and lqn - ql < 612. Letting the desired N equal this No, we note

thatlqm - 4.1 = I(qm - 4) + (q - qn)l 14, - 41 + 14 - 4.1 < 812 + 612 =

E, whenever m, n E N with m, n 2 N, as desired.

DEFINITION 2

A sequence of rational numbers satisfying the property derived in Theorem 1

is called a Cauchy sequence of rational numbers.

Intuitively, Cauchy sequences are sequences whose terms "eventually''

all become closer and closer to one another (in comparison to a convergent

sequence, whose terms eventually all become arbitrarily close to a particular

number). Theorem 1 tells us that any sequence of rational numbers that

converges to a rational number must have this Cauchy property. The ex-

ample given earlier, involving n, provides a clue that there are Cauchy

sequences of rationals that do not converge to rationals. It suggests, fur-

thermore, that if sequences of rationals are going to converge to something

outside Q (something that doesn't exist in our discussion as yet, but that

it is our goal to construct), then the only sequences that "have a prayer"

of doing so are Cauchy sequences.

Hence we are motivated to begin our construction by considering the

collection Ci of all Cauchy sequences of rational numbers. We make O: into

an algebraic structure by defining operations of addition and multiplication

in componentwise fashion. That is, the nth term of the sum (respectively,

product) of two sequences in 6 is the sum (respectively, product) of the

respective nth terms. In order for these operations on 6 to be meaningful,

we need to be certain that the result of combining two elements of Ci is an

element of & (we are, of course, referring here to the issue of closure).

Toward this end, we have Theorem 2.

THEOREM 2

If {an} and {b,} are cauchy sequences of rationals, so are the sequences

{a, + b,}, {anbn} and {-a,). Also, any constant sequence is Cauchy, including

the sequences {O,O, 0,.. .) and (1, 1, 1,.. .}.

We begin here to give only comments on proofs, rather than the proofs
themselves, of most theorems. In reference to Theorem 2 we note that a
necessary lemma for proving closure under multiplication is the fact that
every Cauchy sequence of rationals is bounded in Q [see Exercise l(a)J.
That is, if {q,) is Cauchy, then there exists a rational number B such