The reals form an ordered field

Statement

The real numbers $\mathbb{R}$, with their addition, multiplication and order $\le$, form an ordered field. That is, $\le$ is a total order with two properties for all real numbers $a$, $b$ and $c$. If $a \le b$, then $a + c \le b + c$. If $0 \le a$ and $0 \le b$, then $0 \le a b$.

Remarks

The ordered field $\mathbb{R}$ contains $\mathbb{Q}$ as an ordered subfield. The order of $\mathbb{R}$ is the only order that makes $\mathbb{R}$ an ordered field. The nonnegative reals are exactly the squares.

Explanations

L1 human

All the usual rules for inequalities hold for the real numbers. For example, from $a \le b$ you get $a + 5 \le b + 5$, and from $a \ge 0$ and $b \ge 0$ you get $a b \ge 0$.