# Complex Analysis/Appendix/Proofs/Triangle Inequality

Let and be complex numbers. Since we have:

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

- , and .

Also, the induction shows:

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:

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