Wondering where my logic is going wrong in this assignment:
Show that $||x|-|y|| \leq |x-y|$
Using the fact $||x|-|y||, |x-y| \geq 0$
It follows $(|x|-|y|)^2 \leq (x-y)^2$
Using the fact $|x|^2 = x^2$
$x^2 -2|x||y| +y^2 \leq x^2 -2xy +y^2$
$|xy| \leq xy$
Which I know is not true. Thanks for any input.