I recently came across this question given in some olympiad paper and found it quite accessible, so here is the proof in all its glory after a lemma that will be in use. By the way, all the theorem says is that as long as $p \geqslant 1$ we can approximate any real number to a rational number by some arbitary precision. Lemma: Consider $n+1$ positive real numbers $x_0, x_1, x_2, x_3, \ldots, x_n$ all being in $[0,1)$ then for any $i \neq j$ and $i > j$ it so happens that $$\left | x_j - x_i \right | < \frac{1}{n}$$ for $i,j \in \mathbb{Z}^+$