1.4.2 Assume that John will live forever. He plays a certain game each day. Let Ai be the event that he wins the game on the ith day.

(i) Let B be the event that John will win every game starting on January 1, 2015.

Label the following statements as true or false:

(a) B = liminf A,.

(b) B C liminf A,.

(c) B 2 limsup A,.

(d) B = limsup An.

(ii) Assume now that John starts playing on a Monday. Match the following events C1 through C g with events D1 through Dll:

C1 = John loses infinitely many games.

C2 = When John loses on a Thursday, he wins on the following Sunday.

C3 = John never wins on three consecutive days.

C, = John wins every Wednesday.

Cs = John wins on infinitely many Wednesdays.

c6 = John wins on a Wednesday.

C7 = John never wins on a weekend.

C, = John wins infinitely many games and loses infinitely many games.

C g = If John wins on some day, he never loses on the next day.

D1 D2 j=0 j=0 [A7+4A7(j+1)]. A7j+3. == j=0 D3 D = n=0 n [+6A(+1)] {[][]} Lk=n+1 Ak