begin quote ita"
CHAPTER II [:] The Real Number System [...] §3. THE LEAST UPPER BOUND PROPERTY. "end quote
topic index     contents

LUB5: ARBITRARILY ACCURATE APPROXIMATION OF REALS BY RATIONALS

begin quote ita-2-3-10-pg26 "
If   x, ,   0,   then there exists a rational number   r   such that   .
"end quote

In the quote below, I have changed the wording slightly to something with which I am more comfortable - because there is a big difference between "finding" a number and proving that the number exists. Also, please see below for the justifications, in which I have tried to make explicit all of the information which is implicit in Rosenlicht's proof.

begin quote ita-2-3-10-pg26"
To prove this, [note that by] LUB2 [there exists] a positive integer   N   such that   ,   then [by] LUB4 [there exists] an integer   n   such that   .   Then   ,   so   . "end quote

In the second sentence of Rosenlicht's proof, there are four inequalities and one equality.

The Equality: To get from     to   ,   we subtract   n/N.   We get zero on the left because, by the definition of subtraction, we have added to   n/N   its additive inverse. That the equality still holds follows from the definition of addition as a function from one set to another, which includes the fact that functions have one and only one value for any given argument. For addition, the argument is an ordered pair of real numbers, and for the additions done here the two ordered pairs are     and   .   Because of the equality with which we start, these are one ordered pair.

The Inequalities: To get from     to   ,   again we subtract   n/N,   and this is two applications of O3. Then we append the result of the application of LUB2, and use the definition of the absolute value of a real number.

on to §4: Decimals