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.
Solution 1:
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*} (B^{i}_{n})_{n\in\mathbb{N}}=(A_{nm+i})_{n\in\mathbb{N}}, \quad i=0,1..,m-1 \end{equation*}