$\sum\limits_{n=1}^\infty |a_n|$ converges implies $\sum\limits_{n=1}^\infty |a_n|^2$ converges? [duplicate]
Solution 1:
Hint:
If $\sum |a_n|<\infty$ then $a_n\to 0$
If $a_n\to 0$ then after some $n$ you have $a^2_n\leq |a_n|$ (why?)
Solution 2:
If $\sum |a_n|$ converges, then $|a_n|\to 0$, and thus for sufficiently large $n$, say $n>N$, $|a_n|<1$.
For such $n>N$, $|a_n|^2<|a_n|$, so $\sum |a_n|^2<\sum |a_n|$ and thus by comparison, $\sum|a_n|^2<\infty$.