Application of Borel-Cantelli for sequence of two parameters

Question

Let $(A_{m,\ell})_{\ell \geq 0, m \geq 0}$ be a sequence of events in some probability space. How to show by using Borel Cantelli that, if $$\sum_{\ell \geq 0, m \geq 0} P(A_{m,\ell}) < \infty,$$ then $P( \mbox{number of pairs } (m,\ell) \mbox{ such that } A_{m,\ell} \mbox { occurs} < \infty) =1?$

Answer 1

You have to relabel the elements: use a bijection $\phi$ from $\mathbb N$ to $\mathbb N^2$ and define $B_n:=A_{\phi(n)}$. The assumption gives that $\mathbb P(\limsup_n B_n)=0$ hence $\mathbb P(\mbox{number of } n \mbox{ such that } B_{n} \mbox { occurs} < \infty) =1$.