So this is from Casella Berger 5.38 b)
the question states `` Let $X_{1},...,X_{n}$ be iid with mgf $M_{X}(t)$. Let $S_{n} = \sum X_{i}$ and $\bar{X_{n}}= \frac{S_{n}}{n}$. use the fact that $M_{X}(0)=1, M'_{X}(0)=EX$ to show that if $EX<0$ then there is a constant $0<c<1 s.t. P(S_{n}>a) \leq c^{n}$."
Note that for part a) I showed that $P(S_{n}>a)\leq e^{-at}[M_{X}(t)]^{n} , 0<t<h$.
I'm actually not sure how to show this. Conceptually we know that if the assumption hold and $EX<0$ then the center of the distribution is below zero. using the fact that t=0 $P(S_{n}>a)\leq 1$ is established. not sure how to show more formally however.
Using the assumptions and continuity of $M_X$, we can choose $t\gt 0$ close to $0$ in such a way that $M_X(t)-M_X(0)\leqslant t\mathbb E[X]/2$ (since the derivative is negative on a small interval centered at zero). We thus get that $M_X(t)\leqslant 1+t\mathbb E[X]/2\lt 1$.