In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. For every strictly positive integer $n$, $S_{n}=\sum_{k=1}^{n} X_{k}$ where $\left(X_{k}\right)$ are mutually independent with the same distribution as $X$. Show that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}\right| \geqslant \varepsilon\right) \leqslant 2 \exp\left(-n \frac{\varepsilon^{2}}{2c^{2}}\right)$.
In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. For every strictly positive integer $n$, $S_{n}=\sum_{k=1}^{n} X_{k}$ where $\left(X_{k}\right)$ are mutually independent with the same distribution as $X$.
Show that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}\right| \geqslant \varepsilon\right) \leqslant 2 \exp\left(-n \frac{\varepsilon^{2}}{2c^{2}}\right)$.