TITLE: $\sum\limits_{n=1}^\infty |a_n|$ converges implies $\sum\limits_{n=1}^\infty |a_n|^2$ converges?
QUESTION [13 upvotes]: Possible Duplicate:
Prove that $\sum_{n=1}^{\infty}\ a_n^2$ is convergent if $\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent 

If $\sum\limits_{n=1}^\infty |a_n|$ converges, the $\sum\limits_{n=1}^\infty (a_n)^2$ is also always convergent?

REPLY [7 votes]: 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, $\sum|a_n|^2<\infty$.