 Solution

Interpolation means $p_n(x_i)=f(x_i)$ for every node. Define the Lagrange basis polynomialsDistinctness of the nodes makes every denominator nonzero, and ${\ell }_i(x_j)={\delta }_{ij}$. Thusis a polynomial of degree at most $n$ with the required values. If two such polynomials existed, their difference would have $n+1$ distinct roots while having degree at most $n$; the polynomial root bound makes it identically zero. Hence existence and uniqueness hold. The actual degree can be smaller than $n$, despite the conventional name.