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.