Hello everybody, help with this proof please: proof
The writer say I have to proof this:
|A1 + Aj - M | + |A2n + Ai - M| <= | A1 + A2n - M | + | Aj + Ai - M | but for me is: | A1 + Aj - M | + |A2n + Ai - M | >= | A1 + A2n - M | + | Aj + Ai - M | then I reduced that: | A1 + Aj - M | + A2n + Ai - M >= A1 + A2n - M + | Aj + Ai - M | | A1 + Aj - M | + Ai >= A1 + | Aj + Ai - M | then if A1 + Aj - M >= 0, so Aj + Ai - M >= 0 too, so we have: A1 + Aj - M + Ai >= A1 + Aj + Ai - M, wich is correct. but if A1 + Aj - M < 0, we have to make two cases if Aj + Ai - M < 0, we have : M - A1 - Aj + Ai >= A1 + M - Aj - Ai Ai >= A1, which is correct but if Aj + Ai - M >= 0, we have : M - A1 - Aj + Ai >= A1 + Aj + Ai - M M >= A1 + Aj, wich is not always true. I don't know how to finish this proof, maybe for any condition is not necesary to solve this last case or I was wrong in some case.
For a better understanding this is the problem: problem: