Some of the things I've learned every day since Oct 10, 2016
Category Archives: analysis
January 29, 2017Posted by on
The Dedekind cut is a method of constructing from . A cut is first defined to be a partition of into 2 non-empty subsets such that
- if and , then
- if , then there is a such that (meaning has no maximum element)
A cut can be equivalently determined solely by alone, rather than the pair .
Cuts can then be used to construct by defining any to be the cut where is the set of all members of such that . That is, is simply defined as the subset of rationals smaller than itself, and is the set of all such subsets.
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.
December 8, 2016Posted by on
Where , exists if and only if or .