Integer-valued random variables must converge in distribution to a integer-valued random variable? Let $X_1,X_2,...$ be a sequence of integer-valued random variables that converge in distribution to some random variable $X$. Am I right in thinking that we can always pick $X$ to be integer valued?
I thought like this: Let $F_n$ be the distribution function of $X_n$ and $F$ be the distribution function of $X$. Then $X$ cannot be integer valued only if there are two points $x_1$ and $x_2$ such that
$$F(x_1) \neq F(x_2)$$ 
and
$$|x_1-x_2|<1$$
Now since a distribution function cannot have more than a countable number of points of discontinuity, we can pick $x_1$ and $x_2$ such that they are points of continuity of $F$. But then for large enough $n$
$$F_n(x_1) \neq F_n(x_2)$$
But this contradicts the fact that $X_n$ is integer-valued.
Am I right?
[This is motivated by exercise 5.12 of Wasserman's All of Statistics where he assumes that $X$ is integer-valued]
 A: The CDFs $F_i$ all have the property that $F_i$ is constant on $(n,n+1)$ for every integer $n$. This property is conserved by the limiting function, and is in fact equivalent to the random variable being supported on the integers.
A: There are problems with your proof.

I thought like this: Let $F_n$ be the
  distribution function of $X_n$ and $F$
  be the distribution function of $X$.
  Then $X$ cannot be integer valued only
  if there are two points $x_1$ and
  $x_2$ such that $$F(x_1) \neq F(x_2)$$
  and $$|x_1-x_2|<1$$ (**)

Not necessarily. What if the range of $X$ are non-integers with spaces between them bigger or equal to 1? For example , all the points $n+1/2$, $n\in \mathbb{Z}$ 

Now since a distribution function
  cannot have more than a countable
  number of points of discontinuity, we
  can pick $x_1$ and $x_2$ such that
  they are points of continuity of $F$.

You pick such points of continuity, but at the same time you pick them with the property (**) from the first part. You have to justify that.
A: $X$ has to be an integer valued random variable.
$X_n$ converges to $X$ in distribution if $F_n$ is the cdf of $X_n$ and if $F$ is the cdf of $X$, then $$\lim_{n \rightarrow \infty} F_n(x) = F(x)$$ Note that if $X_n$ is an integer valued random variable then $F_n(x) = F_n(\lfloor x \rfloor)$. 
Consider $|F(x) - F(\lfloor x \rfloor)|$ and argue that this can be made smaller than $\epsilon$, $\forall$ $\epsilon >0$, by adding and subtracting $F_n(x)$ and making use of the fact that $F_n(x) = F_n(\lfloor x \rfloor)$ and then use triangle inequality and choose $N(\epsilon)$ such that both the $\frac{\epsilon}{2}$ are satisfied.
