An amusing sequence of functions

We consider the amusing sequence of functions $f_n: \mathbb{R} \rightarrow \mathbb{R}$ given by $$f_n(x) = \sum_{k=1}^{n}{\frac{|\sin{(k \pi x)}|}{k}}.$$ Every rational point is eventually the location of a strict local minimum of $f_n$: more precisely, $f_n$ has a strict local minimum in all rational points $x=p/q \in \mathbb{Q}$ with $|q| \leq \sqrt{n}$.