How to prove this property of absolute values

Question

Can anyone please help me on how to prove this property for absolute values:

$|a-b|\ge||a|-|b||$

Answer 1

$|a-b|+|b|\ge |(a-b)+b|=|a|$ $\implies$ $|a-b|\ge|a|-|b|$

$|a|+|b-a|\ge|a+(b-a)|=|b|$ $\implies$ $|a|-|b|\ge -|a-b|$

$|a-b|\ge|a|-|b|\ge-|a-b|$ $\implies$ $|a-b|\ge||a|-|b||$

Answer 2

Alternatively: $$|a-b|\cdot |a+b|=||a|-|b||\cdot ||a|+|b|| \Rightarrow |a-b|\ge ||a|-|b||,$$ because: $$|a+b|\le |a|+|b|=||a|+|b||.$$