 Solution

Define the cardinal polynomialsThey satisfy ${\ell }_k(x_i)={\delta }_{ik}$. Hence evaluating the proposed polynomial interpolation formula at $x_i$ gives $p_n(x_i)=a_i$, so the required coefficients areThe polynomial ${\sum }_k{f}_k{\ell }_k$ has degree at most $n$ and takes every prescribed value. Its uniqueness follows because the difference of two such polynomials would have $n+1$ distinct roots while having degree at most $n$.