Proving the Density of a Specific Set in Real Numbers

[D.K.]

Is there an easy way to prove that for any irrational $$\xi$$ the set:

$$\{x \in \mathbb{R}: x = p + q\xi, \ p, q \in \mathbb{Z}\}$$ is dense in $$\mathbb{R}$$?

I know a proof involving notions from measure theory of which I unfortunately know nothing about. Any help would be very appreciated.

 

[D.K.]

Nevermind, it turned out to be rather easy.