Weak convergence of discrete random variables

Question

Let $X_n$ and $X$ be r.v.s taking values in $\mathbb{Z}$. Suppose that $\lim \inf_{n \to \infty} \mathbb{P}(X_n=k) \geq \mathbb{P}(X=k)$ for $k \in \mathbb{Z}$. Show that $X_n$ converges to $X$ in distribution.

We need to show that $F_{X_n}(a) \to F_X(a)$ as $n \to \infty$, I have no idea how to start, any hints would be appreciated.

Answer 1

$X_n$ converges weakly to $X$ if and only if $\liminf P(X_n \in G) \geq P(X\in G)$ for all open sets $G$. Since $X_n$ and $X$ are supported on $\mathbb Z$, the thesis follows.