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.

This inequality is always true. It follows easily by the triangle inequality. 
Start with $$\vert z_1\vert\le\bigl\vert z_1-(-z_2)\bigr\vert+\left\vert-z_2\right\vert\;.$$
