10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 359equivalence class containing the constant sequence with all terms equal to
the integer k) such that [x,] I [k] for all [x,] E X. This k is an upper bound
in R for X; our goal is to produce a least upper bound for X. Such an
object, being an element of R, may be determined by specifying a Cauchy
sequence (4,) of rationals with appropriate properties. The following
procedure can be shown to yield such a sequence. For each positive in-
teger m, let t, be the smallest integer such that the real (in fact, rational)
number determined by the constant sequence {tm/2", t,/2", tm/2",.. .) is
an upper bound for X. The fact that such an integer exists, for each m E N,
depends on the well-ordering principle for N; the existence of the integer
upper bound [k], alluded to earlier, comes into play in this verification.
Next, define a sequence (q,) by letting q, = tn/2" for each n E N; let 1 = [q,].
It can be proved that (q,} is a Cauchy sequence of rationals, so that
I CE R. Finally, it can be shown that 1 satisfies the two conditions required
to conclude that 1 is the least upper bound for the given X. Subject to
rigorous verification of all the details just outlined, we are able to conclude
Theorem 6.
THEOREM 6
The ordered field (R, +, -) is complete.In fact, much more can be proved in reference to the real number field,
including a theorem asserting uniqueness of a complete ordered field. Spe-
cifically, it can be shown that any complete ordered field must be order iso-
morphic to (i.e., for all intents and purposes in an algebraic sense, the same
as) R. As indicated earlier, we will not pursue uniqueness of a complete
ordered field in this text.
Recalling that Q turned out to be the "smallest" field containing Z, you
may wonder whether R has that same relationship to Q. The answer is
"no." The subset Q[*] = (a + bd 1 a, b E Q] is a field (under the usual
operations of addition and multiplication inherited from R) properly con-
tained in R and properly containing Q. The structure of fields such as
Q[-1, known as subfields of R, is commonly studied in the portion of
an introductory abstract algebra course devoted to field theory.
Having fought your way through the outlined construction of R, and
also being familiar with the manner in which the complex number field is
built from R (recall Article 9.4), you may wish to spend a few moments
comparing the two constructions. Such a comparison should remove any
doubt that such descriptive terms as "real" and "imaginary," in reference
to these number systems, are anything other than historical accidents.
Finally, it may be recalled, from Definition 5 and Theorem 1, Article
9.3, that the real number field is Archimedean ordered. Basically, this means
that, given any positive real numbers a and b with a c b, a sufficient num-
ber of additions of a to itself will eventually exceed b. When this concept