Convergence properties of a moment generating function for a random variable without a finite upper bound. I'm stuck on a homework problem which requires me that I prove the following:
Say $X$ is a random variable without a finite upper bound (that is, $F_X(x) < 1$ for all $x \in \mathbb{R}$). Let $M_X(s)$ denote the moment-generating function of $X$, so that:
$$M_X(s) = \mathbb{E}[e^{sX}]$$
then how can I show that
$$\lim_{s\rightarrow\infty} \frac{\log(M_X(s))}{s} = \infty$$

Consider the limit when $s\to+\infty$ of the inequality 
$$
s^{-1}\log M_X(s)\geqslant x+s^{-1}\log(1-F_X(x)).
$$
