Let {An} be a decreasing sequence of events on a sample space, for which we have P(A1) = q, P(An − An+1) = qnp, n = 1, 2,…, where 0 < q < 1 and p = 1 − q.

(i) Show that P(An) = qn for each n = 1, 2,….

(ii) Calculate the probability P(A) for the limiting event A = lim n→∞

An.