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Unformatted text preview: 3. Prove that if A 1 ,A 2 ,... is decreasing then P ( lim n→∞ A n ) = lim n→∞ P ( A n ). Since A 1 ,A 2 ,... is a sequence of decreasing events [hence A c 1 ,A c 2 ,... is a sequence of increasing events] n \ i =1 A i = lim n →∞ A n Deﬁnition of sequence of decreasing events (1) n [ i =1 A c i = lim n →∞ A c n Deﬁnition of sequence of increasing events (2) P ( lim n →∞ A c n ) = lim n →∞ P ( A c n ) Proved in the class (3) 2 Then ( n \ i =1 A i ) c = n [ i =1 A c i P [( n \ i =1 A i ) c ] = P ( n [ i =1 A c i ) P [( n \ i =1 A i ) c ] = 1P ( n \ i =1 A i ) = P ( n [ i =1 A c i ) = P ( lim n →∞ A c n ) = lim n →∞ P ( A c n ) = lim n →∞ (1P ( A n )) 1P ( n \ i =1 A i ) = 1P ( lim n →∞ A n ) = 1lim n →∞ P ( A n ) P ( lim n →∞ A n ) = lim n →∞ P ( A n )...
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 Spring '11
 rsharma

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