Variation of the triangle inequality on arbitrary normed spaces

[Bipolarity]

The following inequality can easily be proved on $ℝ$ :

$||x|-|y|| \leq |x-y|$

I was wondering if it extends to arbitrary normed linear spaces, since I can't seem to prove it using the axioms for linear spaces. (I can however, prove it using the definition of the norm on $ℝ$ by using casework).

Suggestions? Hints?

BiP

 

[fzero]

This is called the reverse triangle inequality and can be shown by applying the regular triangle inequality to

$$ \begin{split} & \| x\| = \| (x - y) + y \|, \\
& \| y\| = \| (y - x) + x \|. \end{split}$$

 

[HallsofIvy]

The "triangle inequality" itself is part of the definition "norm".