= Solution
The <uniqueness theorem for moment-generating functions> states that if $M_X$ and $M_Y$ are finite and equal on an open <real interval> containing zero, then
$$
\boxed{X\overset d=Y.}
$$
For the <convergence theorem for moment-generating functions>, suppose that the <moment-generating functions> $M_{X_n}$ are finite on a common open <real interval> $(-a,a)$, $a>0$, and $M_{X_n}(t)\to M_X(t)$ for every $|t|<a$, where $M_X$ is finite there. Then
$$
\boxed{X_n\overset d\longrightarrow X.}
$$
One may equivalently assume convergence to a finite function $M$ on that interval: the theorem then asserts existence of a unique limiting <probability distribution> with <moment-generating function> $M$ as well as <convergence in distribution>.
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