I have the following exercise:
"For a sequence $(A_k)_{k \geq 1}$ of events, let $N(\omega) = \sum_{k \geq 1} \mathbb{1}_{A_k} ( \omega)$ be the number of $A_k$'s which occur. Show that if $\sum_{k \geq 1} P(A_k) < \infty$, then $P(N < \infty) = 1$. Hint: Show that $\{N = \infty\} \subset B_m$ for all $m \geq 1$, where $B_m = \bigcup_{k \geq m} A_k$."
My issue is, while I can see how to solve the problem using the hint, I can't seem to show the inclusion without using the fact that $\{N= \infty\} = \{A_k \text{ i.o}\}$. But if this was valid, I see no reason for the hint to be there anyway, since we could use the Borel-Cantelli lemma to tell us that $\sum_{k \geq 1} P(A_k) < \infty \ \implies \ P(A_k \text{ i.o} ) = 0$, and then take the complement to get the result.
So my question is, is claiming $\{N= \infty\} = \{A_k \text{ i.o}\}$ too hand-wavey? And if so, what would be the right way of going about showing the inclusion in the hint?
$\{N= \infty\} = \{A_k \text{ i.o}\}$ is true and quite obvious. Nothing hand-wavy about it. But the hint is meant to avoid using Borel-Cantelli Lemma. In fact, the hint is just the way (the easy part of ) Borel-Cantelli Lemma is proved.
[$P(N=\infty)\le \sum\limits_{k=m}^{\infty} P(A_k) \to 0$ as $ m \to \infty$].