Let $X_1,X_2,\ldots$ and $X$ be real-valued random variables. Suppose there exists $a>0$ such that $M_X(t)<\infty$ for every $t \in (-a,a)$ and
\begin{align*}
\lim_{n\to\infty} M_{X_n}(t)=M_X(t)
\end{align*}
for every $t \in (-a,a)$. If each $M_{X_n}(t)$ is finite on $(-a,a)$, then
\begin{align*}
X_n \xrightarrow{d} X.
\end{align*}