Suppose we have measures that defined over the set $\{0,1,2,..,C\}$. Let $\{\mathbb{P}_{n,m}\}$ be a sequence of measures.
Suppose that for fixed $n$, $\mathbb{P}_{n,m}$ converges to $\mathbb{P}_n$ in total variation norm as $m$ approaches infinity. Further, for every $m$, $\mathbb{P}_{n,m}$ converges to $\mathbb{P}$ as $n$ approaches infinity in total variation norm. Then is it true that $\mathbb{P}_n$ converges to $\mathbb{P}$ as $n$ approaches infinity in total variation norm? Please give me a hint. I am struck with this for almost a month.