KöMaL 2013 B4591

Let $\alpha$ be an irrational number. For every positive integer $q$, let $N_q(\alpha)=\min \left\{{\left|\alpha-\frac pq\right|} \colon {p \in \mathbb{Z}}\right\}$, that is, the distance from the closest fraction that can be represented with a denominator of $q$ (not necessarily cancelled to lowest terms). Show that there exists a $k$ such that $\sum\limits_{q=1}^{k} N_q(\alpha)>1$.