i'm stuck on this one:
Prove that || z1|-|z2|| £ |z1-z2|
Thanks!
The usual triangle inequality tells you that $\displaystyle |z_1| = |z_2 + (z_1-z_2)|\leqslant|z_2| + |z_1-z_2|$, and hence $\displaystyle |z_1|-|z_2|\leqslant|z_1-z_2|$. The same inequality with $\displaystyle z_1$ and $\displaystyle z_2$ interchanged gives $\displaystyle |z_2|-|z_1|\leqslant|z_2-z_1| = |z_1-z_2|$. The two inequalities together give $\displaystyle \bigl||z_1|-|z_2|\bigr|\leqslant|z_1-z_2|$.