7. For a positive integer n, let (n) = (2k, i), where i is the remainder when we divide n by 2k, the largest

7. For a positive integer n, let τ (n) = (2k, i), where i is the remainder when we divide n by 2k, the largest possible power of 2. For example, τ (10) = (23, 2), τ (12) = (23, 4), τ (19) = (24, 3), and τ (69) = (26, 5). In an experiment a point is selected at random from [0, 1]. For n ≥ 1, τ (n) = (2k, i), let Xn =    1 if the outcome is in 8 i 2k , i + 1 2k 9 0 otherwise. Show that Xn converges to 0 in probability while it does not converge at any point, let alone almost sure convergence.