Rational Numbers form Subfield of Real Numbers

Theorem
The field $\left({\Q, +, \times, \le}\right)$ of rational numbers forms a subfield of the field of real numbers $\left({\R, +, \times, \le}\right)$.

That is, the field of real numbers $\left({\R, +, \times, \le}\right)$ is an extension of the rational numbers $\left({\Q, +, \times, \le}\right)$.

Proof
Recall that Rational Numbers form Totally Ordered Field.

We need to show that $\Q \subseteq \R$.

Let $x \in \Q$.

From Rational Number is Real Number:
 * $x \in \R$

Thus by definition of subset:
 * $\Q \subseteq \R$

Hence the result.