I'd like to prove\begin{align} \lim\_{n\rightarrow\infty}a\_n=0 \iff \lim\_{n\rightarrow\infty}a\_n\sin(n t)=0 \quad \forall t\in[0,1]\end{align}

Since $\sin$ is bounded one of the implications is trivial. For the other one letting $t=\frac{\pi}{4 l}, l\in\mathbb{N}$ implies $a\_{n\_k}\rightarrow 0$ for all subsequences satisfying $(n\_k)\_{k\in\mathbb{N}}\subset\mathbb{N}\setminus 4 l \mathbb{N}$. Does this lead anywhere?