The Cauchy-Schwarz inequality states that $|u\xb7v|\le |u||v|$

We are able to transform the above inequality so that it also shows us that

$|u+v|\le |u|+|v|$

But I cannot find a way to show that $\mathbf{|}\mathbf{u}\mathbf{-}\mathbf{v}\mathbf{|}\mathbf{\le}\mathbf{|}\mathbf{u}\mathbf{|}\mathbf{+}\mathbf{|}\mathbf{v}\mathbf{|}$

even though I now this has to be true. Any ideas?