Complex Analysis/Appendix/Proofs/Triangle Inequality

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

the triangular inequality follows after taking the square root of both sides. Note here we used the properties:
 * $$|z+w|^2 \,$$
 * $$ = (z+w) \overline {(z+w)} = (z + w)(\bar z + \bar w)$$
 * $$=|z|^2 + z \bar w + \overline { z \bar w } + |w|^2$$
 * $$=|z|^2 + 2\mbox{Re }(z \bar w) + |w|^2$$
 * $$\le |z|^2 + 2|z||w| + |w|^2$$
 * $$= (|z| + |w|)^2 \,$$
 * }
 * $$=|z|^2 + 2\mbox{Re }(z \bar w) + |w|^2$$
 * $$\le |z|^2 + 2|z||w| + |w|^2$$
 * $$= (|z| + |w|)^2 \,$$
 * }
 * $$\le |z|^2 + 2|z||w| + |w|^2$$
 * $$= (|z| + |w|)^2 \,$$
 * }
 * $$= (|z| + |w|)^2 \,$$
 * }
 * $$\mbox{ Re}(z) \le |z|$$, $$|z| = |\bar z|$$ and $$z + \bar z = 2\mbox{Re }(z)$$.

Also, the induction shows:
 * $$\left | \sum_1^n z_k \right | \le \sum_1^n |z_k|$$