Archimedean property

Let x be any real number. Then there exists a natural numberMathworldPlanetmath n such that n>x.

This theorem is known as the Archimedean property of real numbers. It is also sometimes called the axiom of Archimedes, although this name is doubly deceptive: it is neither an axiom (it is rather a consequence of the least upper bound property) nor attributed to Archimedes (in fact, Archimedes credits it to Eudoxus).


Let x be a real number, and let S={a:ax}. If S is empty, let n=1; note that x<n (otherwise 1S).

Assume S is nonempty. Since S has an upper bound, S must have a least upper bound; call it b. Now consider b-1. Since b is the least upper bound, b-1 cannot be an upper bound of S; therefore, there exists some yS such that y>b-1. Let n=y+1; then n>b. But y is a natural, so n must also be a natural. Since n>b, we know nS; since nS, we know n>x. Thus we have a natural greater than x. ∎

Corollary 1.

If x and y are real numbers with x>0, there exists a natural n such that nx>y.


Since x and y are reals, and x0, y/x is a real. By the Archimedean property, we can choose an n such that n>y/x. Then nx>y. ∎

Corollary 2.

If w is a real number greater than 0, there exists a natural n such that 0<1/n<w.


Using Corollary 1, choose n satisfying nw>1. Then 0<1/n<w. ∎

Corollary 3.

If x and y are real numbers with x<y, there exists a rational numberPlanetmathPlanetmathPlanetmath a such that x<a<y.


First examine the case where 0x. Using Corollary 2, find a natural n satisfying 0<1/n<(y-x). Let S={m:m/ny}. By Corollary 1 S is non-empty, so let m0 be the least element of S and let a=(m0-1)/n. Then a<y. Furthermore, since ym0/n, we have y-1/n<a; and x<y-1/n<a. Thus a satisfies x<a<y.

Now examine the case where x<0<y. Take a=0.

Finally consider the case where x<y0. Using the first case, let b be a rational satisfying -y<b<-x. Then let a=-b. ∎

Title Archimedean property
Canonical name ArchimedeanProperty
Date of creation 2013-03-22 13:00:47
Last modified on 2013-03-22 13:00:47
Owner Daume (40)
Last modified by Daume (40)
Numerical id 9
Author Daume (40)
Entry type Theorem
Classification msc 12D99
Synonym axiom of Archimedes
Synonym Archimedean principle
Related topic ArchimedeanSemigroup
Related topic ExistenceOfSquareRootsOfNonNegativeRealNumbers