Thanks, but I already found a solution. the potential function should be the multiplication of the weights and not their sum.
any way, the big O notation means that there is a constant C and and a natural number N, such that for all n>N , C n log(n) is an upper bound
there are n balls of weight 1/n.
an opponent choose each time a subset of balls that each one has weight less than 1. then each ball in this set, its weight is multiplied by 1+\frac{1}{|S|} where S is the set of balls that the opponent chose.
I need to show that for each choice of subsets...
take the Z_2 field and the vector space (Z_2)^2 meaning the vectors are { (0,0), (0,1), (1,0), (1,1) }
any two of the three last vectors are a base for the vector space