Let $X_n$ be positive integer valued random variables. Show $X_n\Rightarrow X_0$, if and only iff, for every $k\geq 0$, $P[X_n=k]\rightarrow P[X_0=k]$.
My work:
$P[X_n=k]\rightarrow P[X_0=k]$ implies $X_n\Rightarrow X_0$ due to Sheffe's Lemma (convergence of densities implies weak convergence).
The other direction is giving me trouble, and I created a counterexample, which is obviously wrong, but I cannot see why. Here it is
Suppose $X_0=x_0$, constant. Let $X_n=x_0+(-1)^n\cdot n^{-1}$. So $X_n$ jumps around $x_0$. We have that $X_n\Rightarrow X_0$, but $P[X_n=x_0]=0$, whereas $P[X_0=x_0]=1$.
As pointed out in the comments, the problem in this attempt of counter-example is that whather the choice of $x_0$, the random variable $X_n$ is not integer valued, at least for $n$ large enough.