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 be a real number, and let . If is empty, let ; note that (otherwise ).
Assume is nonempty. Since has an upper bound, must have a least upper bound; call it . Now consider . Since is the least upper bound, cannot be an upper bound of ; therefore, there exists some such that . Let ; then . But is a natural, so must also be a natural. Since , we know ; since , we know . Thus we have a natural greater than . ∎
If and are real numbers with , there exists a natural such that .
Since and are reals, and , is a real. By the Archimedean property, we can choose an such that . Then . ∎
If is a real number greater than , there exists a natural such that .
Using Corollary 1, choose satisfying . Then . ∎
If and are real numbers with , there exists a rational number such that .
First examine the case where . Using Corollary 2, find a natural satisfying . Let . By Corollary 1 is non-empty, so let be the least element of and let . Then . Furthermore, since , we have ; and . Thus satisfies .
Now examine the case where . Take .
Finally consider the case where . Using the first case, let be a rational satisfying . Then let . ∎
|Date of creation||2013-03-22 13:00:47|
|Last modified on||2013-03-22 13:00:47|
|Last modified by||Daume (40)|
|Synonym||axiom of Archimedes|