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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

$\lim_{s\rightarrow\infty} \frac{\log(M_X(s))}{s} = \infty$

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    You can write your solution as an answer.2012-11-09

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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)). $