Chapter A.1, Problem 24E

To determine

Give the reason justifying the steps in each of the following proof of the triangle inequality.

$\begin{array}{l}{\left|a+b\right|}^2={\left(a+b\right)}^2\\ {\left|a+b\right|}^2\le a^2+2\left|a\right|\left|b\right|+b^2\\ {\left|a+b\right|}^2={\left|a\right|}^2+2\left|a\right|\left|b\right|+{\left|b\right|}^2\\ \left|a+b\right|=\left|a\right|+\left|b\right|\end{array}$