Suppose that $\{ z_n \}$ is a sequence in $\mathbb{C}$ s.t. $z_n = x_n + iy_n$. Let $z=x+iy$. And, let $x_n, y_n, x, y \in \mathbb{R}$.
Prove that $z_n \rightarrow z$ iff $x_n \rightarrow x$ and $y_n \rightarrow y$.
For "$\Rightarrow $" direction, I've seen it from my textbook saying that "if $\{ z_n \}$ is a Cauchy sequence in $\mathbb{C}$, then $\{ x_n \}, \{ y_n \}$ are Cauchy sequences in $\mathbb{R}$." (without any reasoning)
Thus, I wanted to start by showing $\{ x_n \}, \{ y_n \}$ are Cauchy sequences in $\mathbb{R}$, but I am struggling with this one.
To me, it seems right to use the reverse triangle inequality to separate $z_n$ into $|x_n - x_m| $ and $ | y_n - y_m |$, but I'm not sure about how I should proceed from here.
Can I get any hint for this?