How to use triangle inequality to establish Reverse triangle inequality [duplicate]

inequality absolute-value

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 $x = x - y + y$.

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