To Show
$$ | |x| - |y| | \leq |x - y|$$
Easy Way To See It
Remember that the absolute value/norm of the difference of 2 vectors is the distance between them. Rewrite $$| |x| - |y| |$$ as $$d(|x|, |y|)$$ and $$|x-y|$$ as $$d(x,y)$$.
Now we want to show:
$$d(|x|, |y|) \leq d(x, y)$$
But this is obvious. Making both terms the same sign can only put them on the same side of the number line, which can never increase the distance between them.
And this is why I prefer $$d(x, y)$$ to $$|x - y|$$.