# twist on the triangle inequality proof

The usual triangle inequality tells you that $|z_1| = |z_2 + (z_1-z_2)|\leqslant|z_2| + |z_1-z_2|$, and hence $|z_1|-|z_2|\leqslant|z_1-z_2|$. The same inequality with $z_1$ and $z_2$ interchanged gives $|z_2|-|z_1|\leqslant|z_2-z_1| = |z_1-z_2|$. The two inequalities together give $\bigl||z_1|-|z_2|\bigr|\leqslant|z_1-z_2|$.