Rational Numbers form Ordered Field

Theorem
The set of rational numbers $\Q$ forms a totally ordered field under addition and multiplication: $\left({\Q, +, \times, \le}\right)$.

Proof
Recall that by Integers form Ordered Integral Domain, $\left({\Z, +, \times, \le}\right)$ is an ordered integral domain

By Rational Numbers form Field, $\left({\Q, +, \times}\right)$ is a field.

In the formal definition of rational numbers, $\left({\Q, +, \times}\right)$ is the quotient field of $\left({\Z, +, \times, \le}\right)$

By Total Ordering on Quotient Field is Unique, it follows that $\left({\Q, +, \times}\right)$ has a unique total ordering on it that is compatible with $\le$ on $\Z$.

Thus $\left({\Q, +, \times, \le}\right)$ is a totally ordered field.