1. ## supremum

let inf S=m and sup S=M.
if T={|x-y| :x,y belong to S},
then prove that sup T= M-m

2. Originally Posted by luckylawrance
let inf S=m and sup S=M.
if T={|x-y| :x,y belong to S},
then prove that sup T= M-m
What ideas do you have?

3. Intuitively the difference between to elements in S is the biggest when one is the biggest element while the other one is the smallest (or vice versa).

4. Respected, the idea i have is that difference between two numbers is greatest when one is largest and other is smallest in S. but i don't know how to write the proof in proper way.

5. Originally Posted by luckylawrance
Respected, the idea i have is that difference between two numbers is greatest when one is largest and other is smallest in S. but i don't know how to write the proof in proper way.
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 $\displaystyle \alpha-\varepsilon<\alpha$.