Let's suppose that $(a_n)$ is a sequence so that $\sum a_n^2 n^2$ converges, so I have to prove that $\sum |a_n|$ converges.
By the Cauchy criterion we have that, since $\sum a_n^2 n^2 $ converges if $\varepsilon > 0$ then there is a $k\in\mathbb N$ so for all $m,n$ naturals that satisfy $m > n \geq k$: $$ |n^2 a_n^2 + ... + m^2 a_m^2| < \varepsilon $$ but I can't get $| |a_n| + ... + |a_m| | < \varepsilon$ to show that $\sum |a_n|$ converges. Could you give me a hint? Is there any way to prove this without resorting Cauchy criterion. Thanks!