Let $k$ be a strictly positive integer and $U_{1}, \ldots, U_{k}$ a sequence of $k$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed. We also denote
$$S_{k} = \sum_{i=1}^{k} U_{i}$$
Let $\varphi(\lambda) = \ln\left(\mathbb{E}\left[e^{\lambda U_{1}}\right]\right)$.
Let $t \in \mathbb{R}$. Show that for all $\lambda > 0$, we have the inequality
$$\mathbb{P}\left(S_{k} \geqslant t\right) \leqslant \exp(k\varphi(\lambda) - \lambda t).$$