Rational Numbers form Metric Subspace of Real Numbers under Euclidean Metric

Theorem
Let $\struct {\Q, d_\Q}$ be the set of rational numbers under the function $d_\Q: \Q \times \Q \to \R$ defined as:
 * $\forall x, y \in \Q: \map {d_\Q} {x, y} = \size {x - y}$

Let $\struct {\R, d}$ denote the real number line with the usual (Euclidean) metric.

Then $\struct {\Q, d_\Q}$ is a metric subspace of $\struct {\R, d}$.

Proof
From Rational Numbers form Subfield of Real Numbers:
 * $\Q \subseteq \R$

By definition of the Euclidean metric on $\R$:
 * $\forall x, y \in \R: \map d {x, y} = \size {x - y}$

and so $d_\Q$ is a restriction of $d$:
 * $d_\Q = d {\restriction}_{\Q \times \Q}$

From Euclidean Metric on Real Number Line is Metric, $d$ is a metric on $\R$.

The result follows by definition of metric subspace.