How to use triangle inequality to establish Reverse triangle inequality

Question

I need to use $|a+b| \leq |a|+|b|$ to show that $||a|-|b|| \leq |a-b|$ .

I have tried to represent $||a|-|b||$ as $||a|+(-|b|)|$ , and then get $||a|+(-|b|)| \leq |a|+|-|b||$ , but that isn't leading me anywhere given $|a-b| \leq |a|+|b|$.

Answer 1

HINT: supposing $ x \geq y$, consider that $x = x - y + y$.

Answer 2

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.