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):

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