1549901369-Elements_of_Real_Analysis__Denlinger_

1. 7 * "The" Complete Ordered Field 4 7

Lemma 1.7.2 If Fis a complete ordered field, then

Vx E F, x = sup{ r E Qp : r < x}.

That is, in a complete ordered field, every element is the supremum of the

set of all rational elements less than it.

Step 6. Define f: F--+ F' by

f(x) = sup{f(r): r E Qp and r < x}.

(Before you can accept this as a valid definition, you must first prove that
sup{f(r): r E Qp and r < x} exists.)

Step 7. Prove that the function f : F--+ F' just defined satisfies properties

(1), (2), and (9). To do this, you will need results from Exercise Set 1.6-B.

Warning: Completing Step 7 may be quite challenging! •

SUMMARY

Since there is essentially only one complete ordered field, we give it a special
name and symbol. The complete ordered field is called "the Real Number
System" and is denoted with the special symbol, JR..

Definition 1.7.3 (The Real Number System)

JR = the complete ordered field; its members are

called real numbers.

For further insight on the uniqueness of the complete ordered field, the
reader may consult references [10], [15], [37], [63], and [93].