Complex Analysis -Proving convergence Suppose that $$z_n,z\in G:=\mathbb{C}-\{z\,:\,z\leq 0\}$$ and $$z_n=a_n e^{i\theta_n},z=ae^{i\theta}$$ where $-\pi<\theta,\theta_n<\pi$. Prove that if $z_n\to z$ then $\theta_n\to\theta$ and $a_n\to a$.
My attempt:
Since $z_n\to z$. For each $\epsilon>0\,\exists\,m\in\mathbb{Z}^+$ s.t.
$$|z_n-z|=|a_ne^{i\theta_n}-ae^{i\theta}|<\epsilon\quad\forall n\geq m$$
And hints for how to proceed?

$a_n\to a$ follows from
$$|a_n-a|=||z_n|-|z||\leq|z_n-z|<\epsilon$$
and $\theta_n\to\theta$ can be proved from
$$\lim_{n\to\infty}\frac{z_n}{z}=1$$
