Does $\sin(n/x)$ converges a.e. to a constant?

Let us consider the sequence of functions $\sin(n/x)$, for $x\in(0,1)$.

Is it true that there exists a subsequence $n_j$ such that $\sin({n_j}/x)\to 0$ a.e.?

Intuitively, my guess is yes. For every $\varepsilon>0$, fixed $x\in(0,1)$, we can always find a subsequence $n_{j(x)}$ such that $|\sin(n_{j(x)}/x)|\le \varepsilon$. So, one has that $\sin(n_{j(x)}/x)\to 0$. The question is how to avoid the dependence of the subsequence on $x$. My idea is to use a diagonal argument on the point $x$ and the subsequence $j(x)$, but I'm not sure that it works.

Any comments are welcome!