let inf S=m and sup S=M.
if T={|x-y| :x,y belong to S},
then prove that sup T= M-m
Use the characterization that if $\displaystyle A\subseteq\mathbb{R}$ is bounded then $\displaystyle \sup A=\alpha$ if and only if $\displaystyle \alpha$ is an upper bound of $\displaystyle A$ and for every $\displaystyle \varepsilon>0$ there exists $\displaystyle a\in A$ such that $\displaystyle \alpha-\varepsilon<\alpha$.