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? $$\lim_{n\to\infty}\mathbb{P}(\lim_{r\to\infty} \mathcal{A}_{r,n})=0.$$
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?
Again, this isn't necessarily true. Consider the case that $$ \mathbb P(\mathcal A_{r,n}) = \begin{cases} 1 & \text{, if } r > n \\ 0 & \text{, otherwise}. \end{cases} $$