Convergence of Complex Sequence in Polar Form

Theorem
Let $z\neq0$ be a complex number with modulus $r$ and argument $\theta$.

Let $(z_n)$ be a sequence of nonzero complex numbers.

Let $r_n$ be the modulus of $z_n$ and $\theta_n$ be an argument of $z_n$.

Then $z_n$ converges to $z$ iff the following hold:
 * $r_n$ converges to $r$
 * There exists a sequence $(k_n)$ of integers such that $\theta_n+2k_n\pi$ converges to $\theta$.

Proof
Suppose $r_n\to r$ and $\theta_n+2k_n\pi\to\theta$.

We have, by Complex Modulus of Difference of Complex Numbers:

Because $\theta_n+2k_n\pi-\theta\to0$, $\cos(\theta_n+2k_n\pi-\theta)\to1$.

It follows that $|z_n-z|^2\to0$.

Conversely, suppose $z_n\to z$.

By Modulus of Limit, $r_n\to r$.

We have, by Complex Modulus of Difference of Complex Numbers:

By Convergence of Cosine of Sequence, there exists a sequence $(k_n)$ of integers such that $\theta_n+2k_n\pi$ converges to $\theta$.