[Math] Reverse Triangle Inequality with Multiple Terms

Question

Given the reverse triangle inequality:

$$ |x+y| \geq ||x|-|y|| $$

Is it valid to extend this to three terms like so:

$$ |x+y+z| \geq ||x|-|y|-|z|| $$

If this is correct could someone point me in the right direction for a proof? Thanks heaps.

Answer 1

It is false, take $x=y=1$ and $z=-2$. On the other hand $$|x+y+z|=|(x+y)+z| \geq ||x+y|-|z||$$ and by symmetry $$|x+y+z|\geq \max( ||x+y|-|z||, ||x+z|-|y||, ||y+z|-|x||).$$