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 $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. The functions $\Psi$ and $f_{\varepsilon}$ are defined on $\mathbb{R}$, with $f_{\varepsilon}(t) = \mathrm{e}^{-\varepsilon t}\Psi(t)$ (since $m=0$).
Show that $\forall t \in \mathbb{R}^{+*}, f_{\varepsilon}(t) \leqslant \exp\left(-t\varepsilon+\frac{1}{2}c^{2}t^{2}\right)$.