Problem in understanding the converse part of Borel-Cantelli lemma

Hello,

I have read the proof of Borel-Cantelli lemma in Billingzley
(Probability and measure) and Papoulis (Probability,
Random Variables and Stochastic Processes) books.
In the proof of converse Borel-Cantelli lemma, Billingzley has
taken for granted that for proving
A = P( Union (from n=1 to inf) of Intersection (from k=n to
inf) of (A_k complement) = 0
one only needs to prove that for each n:
B = P( Intersection (from k=n to inf) of (A_k complement) = 0

Papoulis used the relation
A <= sum (from n=1 to inf) of B
and then changed the order of limit and sigma. I tried to use
the Lebesgue's dominated convergence theorem to change the order of
lim and sum. But the proof failed.

I appreciate if anyone could give me a hint.

Thanks,
Ehsan
.

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