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 ${}\operatorname {Re} \,{\left(u\right)}\leq \vert {u}\vert$. Therefore,

${}\operatorname {Re} \,{\left(z{\overline {w}}\right)}\leq \vert {z}\vert \vert {w}\vert \,,$

and hence

${}{\begin{aligned}\vert {z+w}\vert ^{2}&=(z+w){\left({\overline {z}}+{\overline {w}}\right)}\\&=z{\overline {z}}+z{\overline {w}}+w{\overline {z}}+w{\overline {w}}\\&=\vert {z}\vert ^{2}+2\operatorname {Re} \,{\left(z{\overline {w}}\right)}+\vert {w}\vert ^{2}\\&\leq \vert {z}\vert ^{2}+2\vert {z}\vert \vert {w}\vert +\vert {w}\vert ^{2}\\&=(\vert {z}\vert +\vert {w}\vert )^{2}.\end{aligned}}$

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

To fact