Hello everyone! I need to solve the following problem but I can't do it.

Let P_{1}and P_{2}be two sigma-additive over the set N of natural numbers probabilities such that P_{1}([0,n])>=P_{2}([0,2]) for all n. I know that for sure there exist n_{1}and n_{2}such that forall c>0, forall n>=n_{2}E_{P2}[(x+c)1_{[0,n]}]>E_{P2}[x] and forall n>=n_{1 }E_{P1}[(x+c)1_{[0,n]}]>E_{P1}[x]. With x a non-negative, real-valued, bounded sequence.

I want to prove that P_{1}([0,n])>=P_{2}([0,2]) implies that whenever for an n E_{P2}[(x+c)1_{[0,n]}]>E_{P2}[x] then for the same n E_{P1}[(x+c)1_{[0,n]}]>E_{P1}[x].

Thanks a lot!