Show that, for all $i$ and $k$ in $\llbracket 1 , n \rrbracket$,
$$L _ { i } \left( a _ { k } \right) = \begin{cases} 1 & \text { if } k = i \\ 0 & \text { otherwise } \end{cases}$$
where $L_i(X) = \prod _ { \substack { j = 1 \\ j \neq i } } ^ { n } \frac { X - a _ { j } } { a _ { i } - a _ { j } }$.