Complex numbers/Modulus/Rules/Fact/Proof

Proof

We only show the triangle inequality, for the other statements see exercise. Because of (7) we have for every complex number $u$ the estimate $\mathrm{Re}\left(u\right)\le |u|$. Therefore,

$\mathrm{Re}\left(z\bar{w}\right)\le |z||w|,$

and hence

$\begin{array}{cc}|z+w{|}^2& =(z+w)(\bar{z}+\bar{w})\\ & =z\bar{z}+z\bar{w}+w\bar{z}+w\bar{w}\\ & =|z{|}^2+2\mathrm{Re}\left(z\bar{w}\right)+|w{|}^2\\ & \le |z{|}^2+2|z||w|+|w{|}^2\\ & =(|z|+|w|{)}^2.\end{array}$

By taking the square root, we get the stated estimate.

To fact