Prove the following
1) ||x|-|y|| is less than or equal to |x-y|.
2) If |x-y| < E for all E > 0, then x = y.
see here
use the contrapositive.2) If |x-y| < E for all E > 0, then x = y.
that is, show that if $\displaystyle x \ne y$, then there is some $\displaystyle \epsilon > 0$ for which $\displaystyle |x - y| \ge \epsilon$
(you need to know the definition of absolute values here)