As in Example 11.3.1, consider the problem of testing P = P0 versus P = P1 based on n i.i.d. observations. The problem is an alternative way to show that a most powerful level α (0 < α < 1) test sequence has limiting power one. If P0 and P1 are distinct, there exists E such that P0(E) = P1(E). Let pˆn denote the proportion of observations in E and construct a level-α test sequence based on pˆn which has power tending to one.