 Solution

Interpolation means $p_n(x_i)=f(x_i)$ for every node. Define the cardinal polynomials ${\ell }_i(x)={\prod }_{j\ne i}(x-x_j)/(x_i-x_j)$, which are well defined because the nodes are distinct. They have degree $n$ and satisfy ${\ell }_i(x_j)={\delta }_{ij}$. Hencebelongs to ${\Pi }_n$ and interpolates. If two polynomials in ${\Pi }_n$ interpolate the same data, their difference has $n+1$ distinct zeros but degree at most $n$, so it is zero. This proves uniqueness. The phrase “of degree $n$” denotes membership in ${\Pi }_n$ here: the actual degree may be smaller, for example for constant data.