Let $X_n$, $n = 1, 2, 3, \ldots$, and $X$ are random variables with at most countably many integer values. Prove that that $X_n \to X$ weakly if and only if $\lim_{n \to \infty} P (X_n = j) = P(X = j)$ for every $j$ in the state space of $X_n$, $n = 1, 2, 3, \ldots$, and $X$.
I managed to prove one direction, i.e. if $X_n$ converges to $X$ weakly than the requirement holds. Would be grateful if you could give an advice how to prove the other direction: assuming $\lim_{n \to \infty} P (X_n = j) = P(X = j)$ for every $j$ in the state space of $X_n$, $n = 1, 2, 3, \ldots$, and $X$, then $X_n \to X$ weakly.
Thanks.