$(a_n)$ 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.

$$V \le a_k \le U\tag{k > 0}$$
$$\frac{1}{k^2}V \le \frac{1}{k^2}a_k \le \frac{1}{k^2}U$$
$$\sum_{k=0}^\infty \frac{1}{k^2}V \le \sum_{k=0}^\infty \frac{1}{k^2}a_k \le \sum_{k=0}^\infty \frac{1}{k^2}U$$
$$\frac{\pi^2}{6}V \le \sum_{k=0}^\infty \frac{1}{k^2}a_k \le \frac{\pi^2}{6}U$$
