# twist on the triangle inequality proof

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• April 2nd 2009, 08:02 PM
morganfor
twist on the triangle inequality proof
i'm stuck on this one:

Prove that || z1|-|z2|| £ |z1-z2|

Thanks!
• April 3rd 2009, 03:42 AM
Opalg
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 $\bigl||z_1|-|z_2|\bigr|\leqslant|z_1-z_2|$.