Prove that $\sum_{n=1}^{\infty}\ a_n^2$ is convergent if $\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent [duplicate]
Solution 1:
$$\lim_{n \to \infty} \frac{a^2_n}{|a_n|} = \lim_{n \to \infty} |a_n| = 0,$$ since $\sum |a_n|$ converges. By the limit comparison test, $\sum a^2_n$ converges.
Solution 2:
Hints:
1) Convergence of an infinite sum implies its terms tend to $0$.
2) If $ |a_n| \le 1$, then $ a_n^2\le |a_n|$.
3) Recall the Comparision Test for infinite sums.