\(\displaystyle{\left|{a}−{b}\right|}\geq{\left|{\left|{a}\right|}−\right|}{b}{\mid}{\mid}\)

\(\displaystyle{\left|{a}\right|}={\left|{a}-{b}+{b}\right|}\le{\left|{a}-{b}\right|}+{\left|{b}\right|}\)

\(\displaystyle{\left|{b}\right|}={\left|{b}-{a}+{a}\right|}\le{\left|{a}-{b}\right|}+{\left|{a}\right|}\)

Thus, we have:

\(\displaystyle-{\left|{a}-{b}\right|}\le{\left|{a}\right|}-{\left|{b}\right|}\le{\left|{a}-{b}\right|}\)