Norm inequality for vectors

Question

I have to show that the following inequality holds for all $a,b\in\mathbb{R}^n\backslash 0$ $$\left\lvert \frac{a}{|a|}-\frac{b}{|b|}\right\rvert\leq \frac{|a-b|}{|a|}+|b|\left\lvert \frac{1}{|a|}-\frac{1}{|b|}\right\rvert.$$ I tried like every identity I know but nothing works, can anyone help me?

Answer 1

Hint: Note that $$ \frac{a}{|a|} - \frac{b}{|b|} = \frac{a}{|a|} + \left(\frac{b}{|a|} - \frac{b}{|a|}\right) - \frac{b}{|b|} = \frac{1}{|a|}(a - b) + \left(\frac 1{|a|} - \frac{1}{|b|} \right)b $$