Real Number Line is Separable

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 1
The rational numbers $\Q$ form a metric space.

We have that the rational numbers are everywhere dense in $\R$.

We also have that the Rational Numbers are Countable.

The result follows from the definition of separable space.

Proof 2

 * Real Number Space is Second-Countable
 * Second-Countable Space is Separable