Complex numbers converge if their absolute values and arguments converge

Question

Let the sequence $\{z_n\}_{n>0}$ and $w \not=0$ be such that $|z_n| \to |w|$ and $\operatorname{Arg}(z_n) \to \operatorname{Arg}(w)$. Show that $z_n \to w$.

My proof: $z_n= |z_n|e^{i \arg(z_n)} \to |w|e^{i \arg(w)}=w$.

Does $\arg(z_n) \to \arg(w)$ if $\operatorname{Arg}(z_n) \to \operatorname{Arg}(w)$?

Answer 1

Hint: $$\left| e^{i Arg(z_n)} -e^{i Arg(w)} \right|=\left| e^{i \left( Arg(z_n)-Arg(w)\right)} -1 \right|$$