Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$.
Show that, for all $i \in \llbracket 0, n \rrbracket$, there exists a unique polynomial $L_i \in \mathbb{R}_n[X]$ such that
$$\forall j \in \llbracket 0, n \rrbracket, \quad L_i(x_j) = \begin{cases} 0 & \text{if } j \neq i, \\ 1 & \text{if } j = i. \end{cases}$$