Complex Analysis/Appendix/Proofs/Triangle Inequality

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

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

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

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

Also, the induction shows:

$\left | \sum_1^n z_k \right | \le \sum_1^n |z_k|$