Real Number Line is Separable/Proof 1

Theorem
Let $\left({\R, \tau_d}\right)$ be the real number line under the Euclidean metric considered as a topological space.

Then $\left({\R, \tau_d}\right)$ is separable.

Proof
The rational numbers $\Q$ form a metric space.

We have that the Rationals are Everywhere Dense in Reals.

We also have that the Rational Numbers are Countably Infinite.

The result follows from the definition of separable space.