Suppose the experiment is to choose a real number at random in the interval (0, 1). For any subinterval (a, b) ⊂ (0, 1), it seems reasonable to assign the probability P[(a, b)] = b−a; i.e., the probability of selecting the point from a subinterval is directly proportional to the length of the subinterval. If this is the case, choose an appropriate sequence of subintervals and use expression (1.3.8) to show that P[{a}] = 0, for all a ∈ (0, 1).