Convergence of Infinite Series

Hello,

Suppose $0<\alpha<\frac{\pi}{2}$ and $\{z_n\}$ is a sequence of complex numbers satisfying $-\alpha<arg(z_n)<\alpha$ . Prove that the two infinite series $\sum_{n=1}^\infty {z_n}$ and $\sum_{n=1}^\infty |{z_n}|$ are both convergent or both divergent.

First, I'm trying to show that if the first converges, then the second also converges. Setting $z_n=x_n+iy_n$ , I know that all $x_n$ are positive and hence $\sum_{n=1}^\infty |x_n|$ converges. But for $y_n$ , this is still unclear for me.

Thank you

Mohammad