Triangle Inequality

Let ${\bf x}$ and ${\bf y}$ be vectors

$\begin{displaymath} \vert{\bf x}\vert-\vert{\bf y}\vert \leq \vert{\bf x}+{\bf y}\vert \leq \vert{\bf x}\vert+\vert{\bf y}\vert. \end{displaymath}$

(1)

Equivalently, for Complex Numbers

and

$\begin{displaymath} \vert z_1\vert-\vert z_2\vert \leq \vert z_1+z_2\vert \leq \vert z_1\vert+\vert z_2\vert. \end{displaymath}$

(2)

A generalization is

$\begin{displaymath} \left\vert{\,\sum_{k=1}^n a_k}\right\vert \leq \sum_{k=1}^n \vert a_k\vert. \end{displaymath}$

(3)

References