Proving the Liouville Numbers are uncountable

The definition I'm given for the Liouville numbers is just that $x$ is Liouville if for every $N > 0$ there exists integers $p,q \geq 2$ such that$$\left|x-\frac{p}{q}\right| < \frac{1}{q^N}.$$

I don't have too many high powered theorems to work with, except for things like the equivalence between Zorn, AoC and Well Ordering, Schroeder-Bernstein and a few other "bread and butter" set theory results. Is there a way to do this with any of those tools, or an explicit bijection between $\mathbb{R}$ (or some other set whose uncountability is easily verified) and the Liouvilles, or something else that uses some of those bread and butter set theory results?

Thanks.