This question already has an answer here:
Show that arbitrarly close to any rational number there is a real (non-rational) number. In other words, show that to each real $\varepsilon>0$ and each rational $r\in\mathbb Q$ there exists $x\in\mathbb R\setminus\mathbb Q$ with $\left|x-r\right|\lt\varepsilon $
No idea how to prove this one. Perhaps I can define some sort of sequence and show it converges...?