===== Order structure of real numbers =====
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $\langle \mathbb R,\le \rangle$ |

We define the total order over the real numbers (in the Dedekind cut model) via $r<s \equiv r\subset s$, i.e.

| @#55EE55: postulate   | @#55EE55: $s \subseteq r \Leftrightarrow s\ge r$ |

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=== Theorems ===
Inequality of arithmetic and geometric means (AM-GM inequality):

^ $\frac{1}{n}\sum_{k=1}^n x_k \ge \left(\prod_{k=1}^n x_k\right)^\frac{1}{n}$ ^

=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Real_number|Real number]], 
[[http://en.wikipedia.org/wiki/Construction_of_the_real_numbers|Construction of the real numbers]], 
[[http://en.wikipedia.org/wiki/Dedekind_cut|Dedekind cut]],
[[https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means|Inequality of arithmetic and geometric means]]

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=== Context ===
[[Real number]]
=== Element of ===
[[Total order]]