Bridge to Abstract Mathematics: Mathematical Proof and Structures

360 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10

was introduced, in Article 9.3, we promised to provide an example of a non-

Archimedean ordered field. Such a field is presented in our concluding

example.

EXAMPLE 1 Consider the algebraic structure REX] of all polynomials in the

single indeterminate x with real coefficients. It can be proved that (R[x],

+, a) is an integral domain (recall terminology introduced in Article

10.2), under the operations of ordinary addition and multiplication of

polynomials. Hence we may use the construction described in Article

10.2 (in building Q from Z) to pass from R[x] to R(x), the field af quo-

tients of R[x], which turns out to be representable as the familiar col-

lection of all rational functions in x with real coefficients. We may

introduce an ordering in R[x] by calling a polynomial positive if the

coefficient of its highest-degree term is positive. This ordering may be

extended to R(x) by declaring a rational function p(x)/q(x) to be posi-

tive in case the product p(x)q(x) is a positive polynomial. This ordering

on R(x) is non-Archimedean, since, for example, no finite number of

additions of the constant polynomial 1 to itself will ever exceed the posi-

tive polynomial x. 0

Exercises

1. This exercise relates to the proof of Theorem 2.

*(a) Prove that every Cauchy sequence of rational numbers is bounded.
(b) Prove that if (a,) and {b,) are Cauchy sequences of rational numbers, then
{a, + b,) and {-a,) are also Cauchy sequences. Prove also that any constant
sequence of rationals is Cauchy.
(c) Use the result in (a) to prove that if {a,) and {b,) are Cauchy sequences of
rationals, then {a,b,) is Cauchy. (Note: For any positive integers m and n,
ambm - a&, = a,bm - a,bm + anbm - a,b,. Use the boundedness of both se-
quences and the triangle inequality.)

2. (a) Prove Theorem 3; that is, prove that relation - defined as 6 is an equiva-
   lence relation.
   (b) Prove Theorem 4(a); that is, prove that addition on R is a well-defined
   operation.
   (c) Prove Theorem 4(b); that is, prove that multiplication on R is well defined.
   (Note: Follow the hints given for Exercise l(c).]


3. (a) Verify, in detail, all field axioms, except Axiom 10, for the structure (R, + , -).
   (b) Prove that the collection of all null sequences of rational numbers constitutes
   an equivalence class in R, by using the two steps:
   (i) Any two null sequences are equivalent.
   (ii) Any sequence equivalent to a null sequence is itself null.
   (c) Prove that the sequence {b,), defined in the verification of Axiom^10 for
   (R, +, -), (cf., Remarks on the proof of Theorem 5), is a Cauchy sequence. (Hint: