Proof for Property of Complex Numbers Is the inequality $\lvert z_1 + z_2 \rvert \ge \lvert z_1 \rvert - \lvert z_2 \rvert$ incorrect, where $z_1$ and $z_2$ are any two complex numbers? I need an example to prove that it is. And in case it is correct, can you please give the proof? Thanks for any help.

We know $\forall z_1,z_2 \in C:|z_1+z_2|\leq |z_1|+|z_2|$ (triangle inequality) 
put $z_1=z_1-z_2 $
$$|(z_1-z_2)+z_2|\leq |(z_1-z_2)|+|z_2|\\|z_1|\leq |z_1-z_2|+|z_2| \to \\
|z_1-z_2|\geq |z_1|-|z_2|$$  Use this fact $ |a||b|=|ab| \to  |z_2|=(1)|z_2|=|-1||z_2|=|-z_2|$ so $$|z_1-(-z_2)|\geq |z_1|-|-z_2| \\\to \\
|z_1+z_2|\geq |z_1|-|z_2|$$
