How to start proof of triangular inequality?

Question

$$\left| {\left| a \right| - \left| b \right|} \right| \le \left| {a \pm b} \right| \le \left| a \right| + \left| b \right| $$

Answer 1

hint: square both sides for both of them and use the facts that: $-|ab| \leq ab \leq |ab|$.

Answer 2

For any $0\le x , y\le 1$ we have $$\color{Red}{-1\le x-y\le 1}$$ Substitute $$x=\dfrac{|a|}{|a+b|}\,\,\,\,\,\text{and},\,\,\,\,\,\,y=\dfrac{|b|}{|a+b|}.$$ Then you will have $$-|a+b|\le |a|-|b|\le |a+b|--------(1).$$ Therefore $$||a|-|b||\le|a+b|-------(2).$$ Replace $b$ by $-b$ in $(2)$. Then $$||a|-|b||\le|a\pm b|.$$ If you replace $a$ by $a-b$ in $(1)$ you can obtain the other side of your inequality.