# Today I Learned

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

## Category Archives: analysis

## 103: Dedekind Cuts

January 29, 2017

Posted 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.

## 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.

## 58: Convergence of Geometric Series with Complex Ratios

December 8, 2016

Posted by on Where , exists if and only if or .

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