Archimedean property
Let x be any real number. Then there exists a natural number 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).
Proof.
Let x be a real number, and let S={a∈ℕ:a≤x}. If S is empty, let n=1; note that x<n (otherwise 1∈S).
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 y∈S 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 n∉S; since n∉S, 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.
Proof.
Since x and y are reals, and x≠0, y/x is a real. By the Archimedean property, we can choose an n∈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.
Proof.
Using Corollary 1, choose n∈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 number a such that x<a<y.
Proof.
First examine the case where 0≤x. Using Corollary 2, find a natural n satisfying 0<1/n<(y-x). Let S={m∈ℕ:m/n≥y}. 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 y≤m0/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<y≤0. 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 |