10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 355that lq,l < B for all n E N. We remark also that once Theorem 2 is known,
it is relatively easy to see that the algebraic structure (6, +,.) satisfies all
except possibly Axiom 10 of the axioms for a field. In fact, it is clear that
Axiom 10 is not satisfied, because (l/n) E 6, while the only possible candi-
date for its multiplicative inverse, namely, in), is not in 6. Sequences con-
taining some (but not all) zero terms also present difficulties, in connection
with Axiom 10. Any such sequence is not the additive identity sequence (all
of whose terms are zero), but its product with any other sequence clearly
will not equal the multiplicative identity sequence (a constant sequence of
l's), and so any such sequence fails to have a multiplicative inverse in 6.
Hence it is clear that the structure (6, + , - ) is not itself a candidate for the
complete ordered field that we seek.
There is another difficulty with the structure 6, yet another reason that
(6, +, .) cannot be the structure we are seeking. This problem, however,
turns out to be part of the solution, since it suggests how we might build
another structure from 6. If we have it in mind to identify irrational num-
bers with Cauchy sequences of rationals that do not converge to rationals
(this, indeed, is our basic premise), then we must note the following. Clearly
there may be many distinct Cauchy sequences that "seem to want" to con-
verge to the same (irrational) number. Consider, for example, our sequence
of decimal approximations to n, described at the outset of this article. We
may easily generate a countably infinite number of different, but "equiv-
alent" Cauchy sequences, one in fact for each positive integer n, by chang-
ing the nth term of the given sequence and leaving all other terms
unchanged. If our original Cauchy sequence seemed to want to converge to
n, then surely all these new sequences, each of which differs from the original
in only a single term, must seem to want to converge to .n also. The answer
to this problem, which amazingly turns out to be the solution to our other
problems as well, is to define an equivalence relation on 6 and to deal with
equivalence classes of Cauchy sequences, rather than with Cauchy sequences
themselves. Denoting by R this collection of equivalence classes of Cauchy
sequences, we announce at this (admittedly premature) stage that this R is
going to turn out to be the underlying set for the real number field! Let us
now proceed with a careful outline of the details.
DEFINITION 3
A sequence {zn}'of rational numbers is said to be a null sequence if and only if to
every positive rational number E, there corresponds a positive integer N such that
1z.I < E whenever n E N with n 2 N.In other words, a null sequence of rationals is simply a sequence of
rationals that converges to zero; the sequence (l/n) is a familiar example.
We now apply the concept of null sequence to aid in formulating the desired
equivalence relation on a. The idea is this: If two sequences of rationals
seem to want to converge to the same number (whether or not rational),