I have the problem:

Prove:

$\displaystyle ||x| - |y|| \le |x - y|$

And I know I'm suppose to use the triangle inequality and the fact:

$\displaystyle x = x + y - y$ and $\displaystyle y = y + x - x$

I tried starting with:

$\displaystyle |x - y| =>$

$\displaystyle |(x - y + y) - (y - x + x)|$

But I don't know how to break this up using the triangle inequality:

$\displaystyle |x + y| \le |x| + |y|$