Series of functions convergence: $a_n \sin\left(\frac 1{tn}\right),\, t\geq 2/\pi$ Problem: If the partial sums of $\sum_{n=1}^{\infty} a_n$ are bounded, then prove that $\sum_{n=1}^{\infty} a_n \sin\left(\frac{1}{tn}\right)$ converges for $t\geq\frac{2}{\pi}$.
I'm trying to use Dirichlet's Test because the partial sums of $\sum_{n=1}^{\infty} a_n$ are bounded and $\lim \sin(\frac{1}{tn})=0$, but I am having trouble showing that $\sin\left(\frac{1}{tn}\right)$ is decreasing for $t\geq\frac{2}{\pi}$. Any help would be much appreciated.

Hint: Show that if $\pi/2 \ge x_1 \ge x_2 \ge \cdots \to 0,$ then $\sin x_n$ decreases to $0.$ Apply this to $x_n = 1/(tn).$
