KöMaL 2013 A603

Let $\alpha$ be a real 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 the sequence $a_k ={\frac{1}{\log k} \sum\limits_{q=1}^{k}} N_q(\alpha)$ is convergent.