I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the moments of the $X_n$ converge to the corresponding moment of $X$, that is, for every $k\ge1$, $$ \lim_{n\to\infty} \mu_k'(X_n) = \mu_k(X). $$ I need to conclude that the $X_n$ converge to $X$ in distribution.

I believe that this is a standard fact in probability, and I would like an excellent source (including a clear statement and proof) for this fact, to cite in a paper I'm writing. (The application is to number theory, which is why I added the probabilistic-number-theory tag.) I also believe that this conclusion holds for many, but not all, random variables $X$ and not just a normally distributed one; I'd be happy for a general statement or one that applies only to a normal variable.

Nominations for a good citing source, anyone?