=+4.5.

(a) Show that lim, P(lim inf, A ,, ( 42) = 0. Hint: Show that lim sup, liminf , A ,, 04% is empty.

Put A* = lim sup ,, A ,, and A, = liminf, A ...

(b) Show that P(A ,, - A*) -> 0 and P(A, - A.) -> 0.

(c) Show that A ,, - A (in the sense that A = A* = A, ) implies P(AA A ,, ) - 0.

(d) Suppose that A ,, converges to A in the weaker sense that P(AAA*) =

P(AAA,) -0 (which implies that P(A* - A, ) =0). Show that P(AAA,)-+0

(which implies that P(A,) - P(A)).