Problem:
Suppose that $(X_i)_{i\in\mathbb{N}^+}$ is a sequence of i.i.d. random variables. For some $n\in\mathbb{N}^+$, let $S_n=\sum_{i=1}^n X_i$. Furthermore, let $a$ be a positive constant, and $h$ be the radius of the interval containing $0$ for which the moment generating function of $X_i$ exists and has continuous derivatives.
Given that $m_X(0)=1$ and $m_X'(0)=\mathbb{E}(X)$, show that if $\mathbb{E}(X)<0$, then there exists $0<c<1$ such that $\mathbb{P}(S_n>a)\le c^n$.
Working:
For all $0<t<h$, we have that $\mathbb{P}(S_n>a)\le e^{-at}[m_X(t)]^n.$
It appears as though both the Intermediate Value Theorem and Mean Value Theorem will be useful, but I'm not entirely sure how to apply them to get the desired result.
Since the function $m'_X$ is continuous, we can find $\delta\lt h$ such that if $|s|\lt 2\delta$, then $m'_X(s)-m'_X(0)\leqslant -\mathbb E(X)/2$ (definition of continuity with $\varepsilon:=-\mathbb E(X)/2\gt 0$). We thus have $$|s|\lt 2\delta\Rightarrow m'_X(s)\leqslant \frac{ \mathbb E(X)}2,$$ hence for $0\lt t\lt\delta$, we have $$\tag{*} m_X(t)-1=\int_0^tm'_X(s)\mathrm ds\leqslant t\mathbb E(X) /2 ,$$ from which it follows $$\mathbb P(S_n\gt a)=\mathbb P\left(e^{tS_n} \gt e^{ta}\right)\leqslant e^{-at}\left(m_X(t)\right)^n\leqslant e^{-at}\left(1+\frac t2\mathbb E(X) \right)^n.$$ Since $e^{-at}\lt 1$ and $0\lt 1+t\mathbb E(X) /2=:c\lt 1$ (by $(*)$ and the fact that $m_X(t)$ is non-negative), we are done.