If A1,A2, . . . is a sequence of events then lim supn→∞An is defined as the set of points that are in an infinite number of the events An,n ≥ 1; and lim infn→∞An is defined as the set of points that are in all but a finite number of the events An,n ≥ 1.

(a) If An,n ≥ 1 is an increasing sequence of events, show that lim sup n→∞

An = lim inf n→∞

An =∪∞

i=1Ai .

(b) If An,n ≥ 1 is a decreasing sequence of events, show that lim sup n→∞
An = lim inf n→∞
An =∩∞
i=1Ai .