Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.3 COMPLETENESS IN AN ORDERED FIELD 313

Solution Suppose there is a rational number u such that u = lubQ S. If
u exists, then either u2 < 3, u2 > 3, or u2 = 3. Now if u2 < 3, the ra-
tional number x = u + [(3 - u2)/7] is clearly greater than u and [it can
be proved-see Exercise 7(a)] satisfies x2 < 3; thus x E S. The facts x E
S and x > u contradict the assumption that u is an upper bound of S.
Hence u2 < 3 must be false. On the other hand, if u2 > 3, we can prove
that the rational number y = (u2 + 3)/2u is less than u and is an upper
bound for S in Q [see Exercise 7(b)-prove this by showing y2 > 31.
This contradicts the assumption that u is a least upper bound for S.
Since it is false that either u2 < 3 or u2 > 3, we must conclude u2 = 3.
But it can be proved, in a manner analogous to the proof in Example
9, Article 6.2, that no rational number satisfies this equation. Our con-
clusion is that S, although nonempty and bounded above in Q, has no
least upper bound in Q. 0

With the result of Example 3, we are on the verge of a precise formulation
of the theoretical difference between Q and R.

DEFINITION 4

An ordered field F is said to be complete if and only if every nonempty subset of F

that is bounded above in F has a least upper bound in F. Otherwise F is said to

be incomplete.

By its definition, the ordered field R is complete. By Example 3, Q is
an incomplete ordered field.
Let us review once again the criteria by which we are now able to
differentiate in an abstract way among the familiar structures N, Z, Q, R,
and C, where each is equipped with the operations of addition and mul-
tiplication. As we've repeatedly stressed, (R, +, .) is, by definition, a com-
plete ordered field (we now know what this means). At this point we assume
that a complete ordered field exists (we pursue this matter further in Article
10.3) and remind you that at most one complete ordered field can exist.
The structure (Q, +, .) of rationals is an ordered field, but it is incomplete.
The complex numbers (C, +, .) constitute a field (see Article 9.4), but a field
that cannot be ordered. The integers (Z, + , -) and positive integers (N, + , -)
each violates one or more of the field axioms; that is, each fails to be a field.

SOME CONSEQUENCES OF THE

COMPLETENESS PROPERTY OF R

A number of important theorems about the real numbers and about func-
tions of a real variable depend on the completeness property of R for their
validity. Suc results are generally stated and used, but not proved, in ele-
mentary and P. intermediate calculus courses. The purpose of the material
that follows is to expose you to the theoretical foundation of several familiar
properties.