In this video (exact time already selected in the link) the connection between so-called 'heavy tails' and an infinite moment generating function is explained as follows:
The benchmark to break into 'heavy' and 'light'-tailed distributions is the exponential. The survival function of an exponential distribution is $\bar F_{\text{exp}}(x) = \Pr(X>x)=e^{-\lambda x}.$ Considering non-negative rv, a distribution is heavy-tailed if:
$$\lim \sup_{x\to\infty} \frac{\bar F(x)}{\bar F_{\text{exp}}(x)}= \lim \sup_{x\to\infty} \frac{\bar F(x)}{e^{-\lambda x}}= \lim \sup_{x\to\infty} \bar F(x)\;\color{red}{e^{\lambda x}}=\infty$$
which is equivalent to saying that the MGF is infinite for all $\lambda>0$.
The presenter points to the part in red (if I understood it correctly), which certainly would blow up with a positive $\lambda,$ but he never connects that part of this expression above with the definition of the MGF, i.e. $\mathbb E(e^{\lambda X}).$
What is the connection between both expressions?
The answer is "promised" in the Wikipedia entry for heavy-tailed distributions by actually defining heavy tails as those distributions with infinity MGF:
\begin{align} M_X(t)=\mathbb E\left[e^{t X}\right]&=\int_{-\infty}^{\infty}e^{tx}f_X(x)dx\\ &=\int_{-\infty}^{\infty}e^{tx}dF_X(x)=\infty \end{align}
and claiming that this implies that
\begin{align} \lim_{x\to\infty}e^{tX}\bar F(x)=\lim_{x\to\infty}e^{tX}\Pr(X>x)=\infty \end{align}
This implication is not proven, and it is my question.