Rationals are Everywhere Dense in Reals

Theorem
The set of rational numbers $\Q$ form a subset of the set of real numbers $\R$ which is everywhere dense.

Proof
From the example given in limit point:

Any point $x \in \R$ is a limit point of the set of rational numbers $\Q$.

This is because for any $\epsilon > 0$, there exists $y \in \Q: y \in \left({x \, . \, . \, x + \epsilon}\right)$ from Between Every Two Reals Exists a Rational.

The result follows from the definition of everywhere dense.