In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real. The function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{tX}\right)$ is defined on $[-\alpha, \alpha]$, and $f_{\varepsilon}(t) = \mathrm{e}^{-(m+\varepsilon)t}\Psi(t)$.
a) Let $t$ be a real belonging to the interval $]0, \alpha]$ and let $n$ belong to $\mathbb{N}^{*}$. Show that $\mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right)=\mathbb{P}\left(\mathrm{e}^{t S_{n}} \geqslant\left(\mathrm{e}^{t(m+\varepsilon)}\right)^{n}\right)$, then that $\mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right) \leqslant\left(f_{\varepsilon}(t)\right)^{n}$.
b) Deduce that there exists a real $r$ belonging to the interval $]0,1[$ such that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right) \leqslant r^{n}$.