Denote points P0, P1, P2, . . . on a circle of radius 1 in the following

way. The point P0 is at the intersection of the x-axis and the circle. The point

P1 is obtained by rotating the point P0 by angle 1 (in some units, for instance

radians) in the counterclockwise direction, point P2 by rotating the point P1 by

angle 1/2 in the counterclockwise direction and so on. In general, Pn+1 is obtained

by rotating Pn by angle 1/(n+1) in the counterclockwise direction. Select a point

Q uniformly at random on this circle. Define the random variable Xn to be the

be the indicator of the event that Q falls in the smaller arc between the points Pn

and Pn+1. Show that the sequence X1,X2, . . . converges to 0 in probability, but

not almost surely.