I'm just checking to make sure I am right about the following (I won't get into detail as the proofs are long and I start to get confused if I look at them too long)
Is it true that $|z_2|-|z_1| \leq |z_1+z_2| \leq ||z_2|-|z_1|| \leq |z_1| + |z_2|$?
I know most (if not all is right) it is mainly the $||z_2|-|z_1|| $ that is throwing me off.
Thank you
Actually it's not true. The triangle inequality says $||z_1|-|z_2||\leq |z_1 + z_2|\leq |z_1| +|z_2|.$ Take $z_1 =2$ and $z_2=1$, then you can have a contradiction for the second inequality you claimed above.