property of the real number
s can be stated as follows: for any real number x
, there exists an integer greater than x
. (In other words, the set of integers is not bounded above
.) A corollary
of this fact, which is itself sometimes called the "Archimedean property", is that for any positive real
, there exists an integer n
such that n x
is greater than y
This property of the reals can be used to prove that there is a rational number between any two reals.
Archimedes' property is a consequence of the completeness axiom (or least upper bound property) for the real numbers, which states that any non-empty subset of the reals which is bounded above has a supremum. A proof of the property runs as follows: assume that there is some real number x such that no integer is greater than x. Then x is an upper bound for the integers, so by the completeness axiom the set of integers has a supremum--call it s. Because s is a least upper bound, s-1 cannot be an upper bound for the integers; hence there is an integer n which is greater than s-1. But then n+1 is an integer greater than s, a contradiction.