
“real: chapter_01” — 2011/5/22 — 12:15 — page7—#7
Basic Properties of the Real Number System 1-7
Proof Since y − x > 0 by the Archimedean property, there exists a
positive integer n such that
n(y − x)>1 or that 1 + nx < ny
By Theorem 1.2 .17 (applied to the real number nx), there is an integer
m such that m ≤ nx < m + 1. Hence, nx < m + 1 ≤ nx + 1 < ny or
that x <(m + 1)/n < y and so r = (m + 1)/n meets the requirement
of the theorem.
The conclusion of the theorem can also be expressed by saying that
the set of all rational numbers
Q is dense in R.
Examples 1.2.16 If both x, y are rationals, then (x + y)/2 is also
rational and clearly lies between x and ...