Rational Numbers form Subfield of Real Numbers

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

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

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.