Let $X_{n},X$ be random variables with values in $k$ in $\mathbb{Z}$ and $F_{X_{n}},F_{X}$ their respective cdf's. Show that under the condition that:
$\forall k\in\mathbb{Z}:\text{lim}_{{n\rightarrow\infty}}P(\{X_{n}=k\})=P(\{X=k\})$
$\text{lim}_{n\rightarrow\infty}F_{X_{n}}=F_{X}$ for all $x\in\mathbb{R}$
The main problem for me is one step where you have to take the limit of a infinite sum wich would be nice if you could draw it in. But can you do this ?
$\text{lim}_{n\rightarrow\infty}\sum_{i\in I} P(\{X_{n}=k\})$
Where $I=\{k\in\mathbb{Z}|k<x\}$