How to use triangle inequality to establish Reverse triangle inequality [duplicate]
The answer is quite easy:
$|a-b|+|b|\geq |a|$
$|b-a|+|a|\geq |b|$
Then $|a-b| \geq \max\{|a|-|b|,|b|-|a|\}=||a|-|b||$.
This argument is quite standard and applies in proving the continuity of norms.
HINT: supposing $ x \geq y$, consider that $x = x - y + y$.