Assume for all finite $r>r_0$ that $$\lim_{n\to\infty}\mathbb{P}(\mathcal{A}_{r,n})=0,$$ where $\mathcal{A}_{r,n}$ is an event on some random variables, and $n$ and $r$ are deterministic variables. How to prove the following? $$\lim_{n,r\to\infty}\mathbb{P}(\mathcal{A}_{r,n})=0.$$
This is a part of a bigger problem. I need to prove the second equation but I have proved the first one. Is the second equation always true? If yes, how can I prove that?
This is not necessarily true. Consider for example the case that
$$ \mathbb P(\mathcal{A}_{r,n}) = \frac{r}{r + n} $$ Then clearly $$ \lim_{n \to \infty} \mathbb P(\mathcal{A}_{r,n}) = 0 $$ for every $r$ but $$ \lim_{r,n \to \infty} \mathbb P(\mathcal{A}_{r,n}) \neq 0, $$ since $$ \lim_{m \to \infty} \mathbb P(\mathcal{A}_{m,m}) = \frac{1}{2}. $$