# Complex Analysis/Appendix/Proofs/Triangle Inequality

Let ${\displaystyle z}$ and ${\displaystyle w}$ be complex numbers. Since we have:
 ${\displaystyle |z+w|^{2}\,}$ ${\displaystyle =(z+w){\overline {(z+w)}}=(z+w)({\bar {z}}+{\bar {w}})}$ ${\displaystyle =|z|^{2}+z{\bar {w}}+{\overline {z{\bar {w}}}}+|w|^{2}}$ ${\displaystyle =|z|^{2}+2{\mbox{Re }}(z{\bar {w}})+|w|^{2}}$ ${\displaystyle \leq |z|^{2}+2|z||w|+|w|^{2}}$ ${\displaystyle =(|z|+|w|)^{2}\,}$
${\displaystyle {\mbox{ Re}}(z)\leq |z|}$, ${\displaystyle |z|=|{\bar {z}}|}$ and ${\displaystyle z+{\bar {z}}=2{\mbox{Re }}(z)}$.
${\displaystyle \left|\sum _{1}^{n}z_{k}\right|\leq \sum _{1}^{n}|z_{k}|}$