Show that if $\sum_{n=1}^{\infty}a_n^2$ converges, then $\sum_{n=1}^{\infty}{a_n \over n}$ converges absolutely [duplicate]
I am trying to show that if the series $\sum_\limits{n=1}^{\infty}a_n^2$ converges, then the series $\sum_\limits{n=1}^{\infty}{a_n \over n}$ converges absolutely. How might I approach this?
Hint:
$$xy \le \frac12(x^2 + y^2)$$
Put $x = \frac1n$ and $y=|a_n|$.
$$\sum_{n\geq 1}\frac{|a_n|}{n}\stackrel{\text{CS}}{\leq}\sqrt{\sum_{n\geq 1}a_n^2 \sum_{n\geq 1}\frac{1}{n^2}} = \frac{\pi}{\sqrt{6}}\left(\sum_{n\geq 1}a_n^2\right)^{1/2}.$$