Triangle Inequality

We denote the absolute value function as $| x |$ where:

| x | = {\begin{cases} x & x \geq 0 \\ - x & x < 0 \end{cases}

This satisfies:

$| a b | = | a | | b |$
$| a + b | \leq | a | + | b |$

for all $a, b \in R$ . You can prove these just considering the cases when $a, b$ are non-negative or not. Note that $| a - b | = | b - a |$ and we denote this value as the distance between $a, b$ . So the triangle inequality says that the distance from any two points is less than the sum of the distances to their origins respectively, or the distance to some intermediary point $c$ where $d (a, b) + d (b, c) \leq d (a, c)$ .

References

[[Abbott Real Analysis.pdf#page=21]]