## Order structure of real numbers

### Set

definiendum | $\langle \mathbb R,\le \rangle$ |

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

postulate | $s \subseteq r \Leftrightarrow s\ge r$ |

#### Theorems

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

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

Wikipedia: Real number, Construction of the real numbers, Dedekind cut, Inequality of arithmetic and geometric means