Show an infinite series of complex number converges Suppose a sequence of complex number $(z_n)_{n=1}^\infty$. Show that if the series ( of positive numbers) $\sum_{n=1}^\infty {|z_n|^2}$ converges, then the series (of positive numbers) $\sum_{n=1}^\infty \frac{|z_n|}{n}$ also converges.
My attempt: Let $z_n=x_n+iy_n$ where $x_n,y_n \in \mathbb{R}$ for all $n \geq 1$. Notice that $|z_n|^2=x_n^2 +y_n^2$ and also $$\frac{|z_n|}{n}=\frac{\sqrt{x_n^2+y_n^2}}{n} \leq \frac{x_n^2 +y_n^2}{n}\leq x_n^2 +y_n^2$$ for $n \geq 1$. By comparison test, the series $\sum_{n=1}^\infty \frac{|z_n|}{n}$ converges. Is my proof correct?

(to update my comment to an answer)
By Cauch-Schwarz inequality $(\sum |z_n|/n)^2\leq (\sum |z_n|^2)(\sum 1/n^2)$. (use it for sums form 1 to $N$ and take $N$ to $\infty$, or use it for infinite sums, if you know it also in this form)
