Convergence of Complex Sequence Let $z_n = \left(\frac{1+i}{3}\right)^n$ be a complex sequence. Show that $(z_n)$ converges.
I'm unsure how to do this because I've only just started learning about complex sequences. If this were real and, say we replaced $i$ with just $1$, I would note that $2^n < 3^n$ for all $n \geq 1$.
That said, I wonder if it's okay to use the absolute value of the complex number in the numerator and note that $(\sqrt{2})^n < 3^n$ to see that the sequence is convergent, or is this totally wrong?

Yes, the key here is that
$$
\left|
{1 + i \over 3}
\right|
\leq {2 \over 3} < 1.
$$
