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: