1549901369-Elements_of_Real_Analysis__Denlinger_

46 Chapter 1 • The Real Number System

(4) f(lp) = lpt;

(5) Vx E F, f(-x) = -f(x);

(6) Vx E F, f(x-^1 ) = f(x)-^1 ;

(7) For each natural number np E Np, and the corresponding natural number

nF' E NF', f(np) = nF';

(8) For each rational number np E Qp, and the corresponding rational num-

mp

ber --np1 E QF', f ( -np) = --np' ;

mp1 mp mF'

(9) .Vx, y E F, x < y {:::} f(x) < f(y);

(10) For all A ~ F,

(a) A is bounded above in F {:::}the set f(A) = {f(x) : x EA} is bounded

above in F'. Moreover, f(supA) = supf(A);

(b) A is bounded below in F {:::} the set f(A) = {f(x) : x EA} is bounded

below in F'. Moreover, f(inf A)= inf f(A).

*To prove this theorem is beyond the scope of this book. However, it is not
too difficult, and would make a rewarding project for a n enterprising student. A
few remarks about how to proceed with the proof will suffice for our purposes.
First, there is much redundancy in the statement of the theorem. Properties
(3)-(8) are easily derived from properties (1) and (2), and property (10) is
derivable from (9). Thus, all we need to do is show how to construct a 1-1
correspondence f: F---> F' satisfying properties (1), (2), and (9).
Step 1. Define f: Np---> NF' by f(np) = nF'.
Step 2. Extend f to f : 'll,p ---> ZF' by

{

f(np) = nF', Vnp E Np

f(Op) = OF'

f(-np) = -nF', Vnp E Np

Step 3. Extend f to f : Qp ---> QF' by

f (mp)= mp'.
np nF'
Actually, we could have started here, with this as our basic definition. Steps
1 and 2 were written down merely to help conceptualize the process.
Step 4. Prove that this f: Qp---> Qp1 satisfies properties (1), (2), and (9).
Step 5. Prove the following: