$\int_{-\infty}^{\infty}e^{tx}dF(x)=\infty \text{ for all } t>0 \Leftrightarrow \lim_{x \rightarrow \infty}e^{tx}\mathbb{P}[X>x]= \infty \text{ for all }t >0$.
I proved the $\Leftarrow $ direction directly using Chernoff bound but I am unable to prove the $\Rightarrow $ direction. Any idea or hint? Thank you!
A sketch proof of $\implies$'s contrapositive:
By L'Hôpital's rule, $\lim_{x\to\infty}\frac{\Bbb P(X>x)}{e^{-tx}}=\lim_{x\to\infty}\frac{F^\prime(x)}{te^{-tx}}$, so if this is finite for some $t>0$, $\int_{\Bbb R}e^{tx/2}F^\prime(x)dx$ is also finite.