Convergence theorem for moment-generating functions

Suppose the moment-generating functions $M_n$ are finite on a common open interval $(-a,a)$ containing zero and converge there pointwise to the moment-generating function $M_X$ of a random variable $X$, with $M_X$ finite there. Then $X_n$ tends to $X$ in distribution. Equivalently, a finite pointwise limit on such an interval is the moment-generating function of a unique limiting probability distribution. This theorem requires control on a neighborhood of zero; convergence of isolated moments is a different hypothesis.