Convergence of Complex Sequence in Polar Form

Theorem
Let $z \ne 0$ be a complex number with modulus $r$ and argument $\theta$.

Let $\sequence {z_n}$ 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 $z$ the following hold:
 * $(1): \quad r_n$ converges to $r$
 * $(2): \quad$ There exists a sequence $\sequence {k_n}$ of integers such that $\theta_n + 2 k_n \pi$ converges to $\theta$.

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

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

Because Cosine Function is Continuous:
 * $\map \cos {\theta_n + 2 k_n \pi - \theta} \to 1$

It follows that:
 * $\cmod {z_n - z}^2 \to 0$

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 $\sequence {k_n}$ of integers such that $\theta_n + 2 k_n \pi$ converges to $\theta$.