## 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 $$|x - y|$$.