Complex Analysis/Appendix/Proofs/Triangle Inequality

Let $z$ and $w$ be complex numbers. Since we have:

$|z+w|^{2}\,$	$=(z+w){\overline {(z+w)}}=(z+w)({\bar {z}}+{\bar {w}})$
	$=|z|^{2}+z{\bar {w}}+{\overline {z{\bar {w}}}}+|w|^{2}$
	$=|z|^{2}+2{\mbox{Re }}(z{\bar {w}})+|w|^{2}$
	$\leq |z|^{2}+2|z||w|+|w|^{2}$
	$=(|z|+|w|)^{2}\,$

the triangular inequality follows after taking the square root of both sides. Note here we used the properties:

${\mbox{ Re}}(z)\leq |z|$, $|z|=|{\bar {z}}|$ and $z+{\bar {z}}=2{\mbox{Re }}(z)$.

Also, the induction shows:

$\left|\sum _{1}^{n}z_{k}\right|\leq \sum _{1}^{n}|z_{k}|$