On the Wikipedia page they write the set of all Liouville numbers as $\displaystyle{L = \bigcap _ { n = 1 } ^ { \infty } U _ { n }}$, and it's stated that each $U_n$ is an open dense subsets of $\mathbb{R}$, where:
$$U _ { n } = \bigcup _ { q = 2 } ^ { \infty } \bigcup _ { p = - \infty } ^ { \infty } \left\{ x \in \mathbb { R } : 0 < \left| x - \frac { p } { q } \right| < \frac { 1 } { q ^ { n } } \right\} = \bigcup _ { q = 2 } ^ { \infty } \bigcup _ { p = - \infty } ^ { \infty } \left( \frac { p } { q } - \frac { 1 } { q ^ { n } } , \frac { p } { q } + \frac { 1 } { q ^ { n } } \right) \backslash \left\{ \frac { p } { q } \right\}$$
My question: why is each $U_n$ an open dense subset of $\mathbb{R}$?
They're dense because $\overline{U_n} \supset \mathbb{Q}$, so $\overline{\overline{U_n}} = \overline{U_n} \supset \overline{\mathbb{Q}} = \mathbb{R}$.