358 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10
the real number [qb] if and only if (q, - qb) is a null sequence; that is,
(q,) is equivalent to (4;). Noting that any rational q is surely associated
with the real number [q] = [(q, q, 4,.. .)I, we conclude that each rational
number is associated with a unique equivalence class in R, so that the
mapping q -t [q] is well defined. You should be familiar by now with the
litany of details that should be verified, in reference to the mapping q -, [q].
This mapping is one to one, not onto, and preserves addition and multipli-
cation; its image is a subset of R that is closed under addition and multi-
plication (see Exercise 4).
We deal next with the matter of ordering. In order that (R, +, .) be
established to be an ordered field, we must specify the subset of R that will
serve as 9, the positive part of R. It turns out that the definition of 9 is
not especially intuitive. Recall, from the discussion of the proof of Theorem
5, that any nonnull Cauchy sequence has the property that the set of all
terms of the sequence, after some specific term, is "bounded away" from
zero by some specific positive rational number (el2 in the context of that
discussion). More formally, if (a,) is a nonnull Cauchy sequence of ratio-
nals, then there exists a positive rational number E and a positive integer
N such that la,l >_ E for all n > N. This provides the idea for the correct
definition of positivity in R.
DEFINITION 6
An element [a,] of R is said to be positive if and only if there exists a positive
rational number E and a positive integer N such that a, 2 E for all n 2 N.As usual, we must worry about well-definedness. Specifically, we should
verify that if (a,) and (a:) are equivalent Cauchy sequences, then (a,) satis-
fies the property in Definition 6 if and only if {a;) does (see Exercise 5).
Following this, we need to verify that the set 9 of all positive elements
satisfies the requirements of Definition 1, Article 9.2. This done, the con-
clusion that (R, +, .) is an ordered field is justified.
The final step is the verification of completeness. This is technically the
most complicated part of the entire development, and we continue here to
omit most of the details of the proof. Let us, however, look for a moment
at the main lines of the argument. As we have learned to do, we begin by
asking ourselves, "What must we prove?" The answer to this question, of
course, is contained in the definition of completeness. We must let X be a
nonempty subset of R that is bounded above in R. We must show that X
has a least upper bound in R; that is, we must prove existence of a real
number 1 satisfying the two defining properties of the least upper bound
in an ordered field (recall Definition 4, Article 9.3). Now the fact that X is
bounded above in R means that there exists a real number [a,] such that
every element [x,] of X satisfies [x,] [a,]; that is, [a, - x,] is positive in
the sense of Definition 6. It follows that there is an integer [k] (i.e., the