=+E[ f(X )] > 1 as n -> co, so that u(i) = 1. Thus the support set is M = (0), and for an initial state i > 0 the probability of ever hitting M is fro < 1.

For an arbitrary finite stopping time T, choose n so that P.[ 0.

Then E[f(X,)] ≤1- fi+ .... PAT