# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 97: The Archimedean Property of the Reals

January 20, 2017

Posted by on has the **Archimedean property**, which states that for any positive there exists an such that

.

Intuitively, this means that contains neither infinitely large nor infinitesimally small numbers. (The property would not hold if was infinite, or if was infinitesimal.)

#### Proof:

Finding an such that is equivalent to finding one such that . If there is no such , this means that for all and thus the number is an upper bound on . However, has no upper bound, so this is a contradiction, meaning such an must exist.

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