In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as: $$ \begin{equation} M_n (t) = \frac{1}{n}\sum\limits_{i = 1}^{n} e^{t X_i} = \int\limits_{-\infty}^{\infty} e^{tx} dF_n (x), \end{equation} $$ where $F_n (x)$ denotes the empirical distribution function. He then states, without proof, the following proposition:
Let $M(t)$ be the moment generating function of $X$ and assume that $M$ is defined for all $t$ in a non-degenerate interval $J$ then: \begin{equation} \sup_{t \hspace{1mm} \in \hspace{1mm} J} | M_n (t) - M(t) | \to 0, \quad \textrm{as } \quad n \to \infty. \end{equation}
He establishes that the proof can be done by noticing that: \begin{equation} M_n(t) - M(t) = \int\limits_{-\infty}^{\infty} e^{tx} d\big(F_n (x) - F(x)\big), \end{equation} and then dividing the integral "into two parts $\int\limits_{|x| > A} + \int\limits_{|x| \leq A}$ and making use of the Glivenko–Cantelli theorem and another classical result, Dini's theorem" ($A$ is never specified in the article).
I am not interested in other proofs of this (such as the one in p. 459 here that uses convexity) but in understanding how Csörgö did it. Could someone please provide a more detailed guideline as how Csörgö's proof would go?
Thanks!