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

Here is the problem.

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.

#### Solutions Collecting From Web of "Choosing the correct subsequence of events s.t. sum of probabilities of events diverge"

Hint It is not generally true that $\sum P(A_{nm})=+\infty$. But it must be true for one of the $m$ following sequences:
\begin{equation*}