42 Chapter 1 • The Real Number System
But we also know from Theorem 1.2.8 (e) that
(u-8)^2 <u^2 < (u+o)^2.
Adding these two inequalities together, we have
(tl - 8)^2 - ( u + 8)^2 < u^2 - 2 < (u + 8)^2 - (u - 8)^2.
-4uo < u^2 - 2 < 4uo
lu^2 - 21<4uo
lu^2 - 21 < 4u · ~ (since o < ~)
4u 4u
lu2 - 21 < €.
Since this holds 'Ve > 0, we have u^2 = 2 by the "forcing principle." •Corollary 1.6.11 The ordered field Q of rational numbers is not complete.
Proof. By Theorem 1.4.5, Q has no element whose square is 2. By Theo-
rem 1.6.10, a complete ordered field must have an element whose square is 2.
Therefore, Q cannot be complete. •
The completeness property makes no mention of nonempty sets with lower
bounds. The curious student will ask whether, in a complete ordered field, such
sets must have greatest lower bounds. The next theorem answers that question.
Theorem 1.6.12 In any complete ordered field, every nonempty set that has
a lower bou.nd in F has a greatest lower bound in F.
Proof. Exercise 1. •THE REAL NUMBER SYSTEM, JR
It turns out that there is one, and essentially only one, complete ordered
field. This is a deep and fundamental r esult in the foundations of mathematics,
and we shall discuss it briefly in Section 1.7. Meanwhile, we take advantage of
this result to define the real number system.
Definition 1.6.13 The real number system is the complete ordered
field. It is denoted JR. Its elements are called real numbers.
Finally, we admit the symbols +oo and -oo into our system, to be used
for the supremum and infimum of unbounded sets.
SETS WITH NO INFIMUM OR NO SUPREMUM;
THE SYMBOLS -oo AND +ooIn view of Definition 1.6.8 and Theorem 1.6.12, the only sets in a complete
ordered field that have no infimum or no supremum are sets that either are
empty, have no lower bounds, or have no upper bounds. Accordingly, we make
the following: