I think I can prove it by Abel's test, because $\sum\frac{\sin k}k$ is convergent (can be proved by Dirichlet's test) and $k\sin(1/k)$ is monotone and bounded. But to prove that $k\sin(1/k)$ is monotone, I need to use derivative, which my analysis course hasn't touched.
Is there any way to prove it by relying on simple inequalities, such as $\sin x\le x$, etc.?