Geometric Interpretation of $|z_1-z_2|\ge ||z_1|-|z_2||$ I have to give a geometric argument that, given two complex numbers $z_1, z_2$, the following inequality holds
$$|z_1-z_2|\ge ||z_1|-|z_2||$$
I know every complex number has a nonnegative modulus, and this becomes a problem if $|z_1|\lt |z_2|$, and it contradicts the fact that $|z|\ge 0$ for all $z \in \mathbb{C}$.  Intuitively I understand what is happening, but I'm having trouble formulating a geometric argument.  I would imagine that, looking at a Argand diagram, the Triangle Inequality would be
$$|z_1|+|-z_2|\ge |z_1-z_2|$$
which makes sense.  But I'm not sure where to go from here.

If you want to proof it using the triangle inequality:
Apply the triangle inequality to get $|z_1-z_2|+|z_2| \geq |z_1|$ if $z_2\geq z_1$.
Apply it again to get $|z_2-z_1|+|z_1| \geq |z_2|$ if $z_1\geq z_2$.
This gives together with $|x|=|-x|$ and $|x|=x$ if $x \in \mathbb R_{\geq 0}$  the desired inequality. 
