Choosing the correct subsequence of events s.t. sum of probabilities of events diverge

Here is the problem.

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I tried choosing $B_n = A_{mn}$ since it is an independent sequence for $m \geq 2$, but I am not quite sure how to guarantee that $\sum_{n=1}^{\infty} P(A_{mn}) = \infty$.

Is it true? If so, why? If not, what other subsequence can you suggest?

This is from Rosenthal, btw.

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