The definition of dense I'm using is the following: a subset $S$ of $(0, 1)$ is dense in $(0, 1)$ if for every $x \in (0, 1)$ and every $\epsilon > 0$ there exists $s \in S$ with $|s - x| < \epsilon$.
I'm trying to show that the set of irrationals in $(0, 1)$ (i.e. the set $\{x \in \mathbb{R} \setminus \mathbb{Q} : 0 < x< 1\}$ is dense in $(0, 1)$.
My idea is this: given $x \in (0, 1)$ and $N \in \mathbb{N}$, try to find an irrational number $\alpha \in (0, 1)$ that depends on $N$ and $x$ such that $|\alpha - x| < 1/N$. I'm not sure how to construct such an $\alpha$ though. I'm thinking I need to to take some irrational multiple of $\lfloor{Nx\rfloor}$ or something but I can't quite work it out. Can anyone help?