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: