## 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 \(\|x - y\|\).