$(a_n)$ is bounded $\implies \sum a_n \frac{1}{n^2}$ converges Show that if $(a_n)$ is a bounded sequence, then
$$\sum \frac{1}{n^2} a_n $$ converges
I only could prove that this sequence is bounded, but not its convergence.
Thanks in advance.

Given that $|a_n|\le M$, then we can use the comparison test to show that because
$$
\left|\,\frac{a_n}{n^2}\,\right|\le\frac M{n^2}
$$
and that $\displaystyle\sum_{n=1}^\infty\frac1{n^2}$ converges, we also have that $\displaystyle\sum_{n=1}^\infty\frac{a_n}{n^2}$ converges absolutely.
