I'm working on a question I appreciate if you can guide me on how to solve it.
Consider $X_j$ as integer random variables.
We know $Pr(X_j = k) \rightarrow Pr(X = k)$ for every integer k. We also know that X is an integer-valued random variable.
I'd like to show:
$\sum_{k = -\infty}^{\infty} \max\{Pr(X = k) - Pr(X_j = k), 0\} \rightarrow 0$, as $j \rightarrow \infty$
Thanks for your help in advance.