Rational Numbers form Subfield of Real Numbers

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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.

$\blacksquare$


Sources