# Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges. [duplicate]

Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges.

Attempt:

$\sum |a_n|$ converges $\implies \sum |a_n|<M$.

If $\sum |a_n|$ converges, then $\sum a_n$ also converges.

Could anyone please tell me on how to proceed ahead with regard to this problem.

Thank you very much for your help

## marked as duplicate by Git Gud, copper.hat, leo, Lord_Farin, user98602 Jan 26 '15 at 18:18

• $\sum a_n^2 \le (\sum |a_n|)^2$,bounded monotone convergence thm – r9m Jan 26 '15 at 17:48
• After a while $|a_n|\lt 1$, so $a_n^2\lt |a_n|$. – André Nicolas Jan 26 '15 at 17:49
• Note that $|a_n|$ must be bounded by some $M$ and so $|a_n|^2 \le M |a_n|$. – copper.hat Jan 26 '15 at 17:52
Prove that there is a tail of the summation for which $a_n^2<|a_n|$ holds for all terms in the tail.