1549901369-Elements_of_Real_Analysis__Denlinger_

1.6 The Completeness Property 35

where ao, ai,. · ·an are (constant) real numbers. If an -/=-0, then n is

called the "degree" of p( x), and the coefficient an is called the "leading

coefficient" of p(x). A function of the form R(x) = :~:~, where p(x)

and q(x) are polynomials and q(x) is not the zero polynomial, is called a

rational expression (in x).

(a) Let F denote the set of all rational expressions in x, together with

the usual operations of addition and multiplication of rational ex-

pressions. Prove that F is a field.

(b) Let P denote the set of all rational expressions :~:~, where the

leading coefficients of p(x) and q(x) are either both positive or both

negative. Prove that P satisfies the order axioms (01)- (03), and

thus F, together with P, is an ordered field.^8

(c) Identify the "natural numbers" in this ordered field.

(d) Prove that there are rational expressions in F larger than all the

natural numbers in F, and thus prove that this ordered field is non-

Archimedean.

1.6 The Completeness Property

The final defining characteristic of the real number system is that it is a "com-
plete" ordered field. The completeness property is the most difficult of all the
properties to describe. To do so, we first need to make some remarks and ob-
servations about "bounded" sets, and then define "suprema" and "infima" of
sets.

BOUNDED SETS IN ORDERED FIELDS

Definition 1.6.1 Suppose that F is an ordered field, A <;;; F, and u E F. We
say that

(1) u is an upper bound for A if \:/x EA, x ~ u.

(2) u is a lower bound for A if \:/x E A, x ~ u.

(3) u is a maximum (or greatest) element for A if u E A, and \:/x E A,

x ~ u. The notation we use to express this is u = max A.

( 4) u is a minimum (or least) element for A if u E A, and \:/x E A, x ~ u.

The notation we use to express this is u = min A.

8. See Definition 1.2.1.