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:

$\forall x, y \in \Q: \map d {x, y} = \size {x - y}$

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

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

Proof

From Rational Numbers form Subfield of Real Numbers:

$\Q \subseteq \R$

By the 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.

$\blacksquare$

Sources

• 1962: Bert Mendelson: Introduction to Topology ... (previous) ... (next): $\S 2.7$: Subspaces and Equivalence of Metric Spaces: Example $1$