$\sum\limits_{n=1}^\infty |a_n|$ converges implies $\sum\limits_{n=1}^\infty |a_n|^2$ converges?

Question

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?

Answer 1

Hint:

1. If $\sum |a_n|<\infty$ then $a_n\to 0$

2. If $a_n\to 0$ then after some $n$ you have $a^2_n\leq |a_n|$ (why?)

Answer 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, $\sum|a_n|^2<\infty$.