Some of the things I've learned every day since Oct 10, 2016
97: The Archimedean Property of the Reals
January 20, 2017Posted 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.)
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.