Let S be a given sample space and let A1, A2, . . . be an infinite sequence of events. For n = 1, 2, . . . , let Ai = n Ai and let Cn = Ai.
a. Show that B1 ⊃ B2 ⊃ . . . and that C1 ⊂ C2 ⊂ . . . .
b. Show that an outcome in S belongs to the event Ai if and only if it belongs to an infinite number of the events A1, A2, . . . .
c. Show that an outcome in S belongs to the even Cn if and only if it belongs to all the events A1, A2, . . . except possibly a finite number of those events.