02 - Triangle Inequality

Triangle Inequality

Take $a,b,c\in \mathbb{R}$

1. $|a+b|\le |a|+|b|$

2. $|a+b|\ge ||a|-|b||$
   Also called the reverse triangle inequality

Proof

1. As $-|a|\le a\le |a|$ and $-|b|\le b\le |b|$, by Properties of Modulus (5)
   Adding them together then

$$-(|a|+|b|)\le a+b\le |a|+|b|$$

Hence by Properties of Modulus (6) then

$$|a+b|\le |a|+|b|$$

2. By $(1)$ then

$$\begin{array}{rl}|a|& =|(a+b)+(-b)|\le |a+b|+|-b|=|a+b|+|b|\\ |b|& =|(b+a)+(-a)|\le |a+b|+|-a|=|a+b|+|a|\end{array}$$

Hence $|a|-|b|\le |a+b|$ and $|b|-|a|\le |a+b|$
Hence by Properties of Modulus (5)

$$|a+b|\ge ||a|-|b||$$