 Solution

The Lagrange interpolation polynomial constructs existence explicitly:Distinct nodes make every denominator nonzero, and ${\ell }_j(x_k)={\delta }_{jk}$, so $p(x_k)=f_k$ and $p$ has degree at most $n$. If two such polynomials existed, their difference would have degree at most $n$ but at least $n+1$ distinct roots. The root bound for a nonzero polynomial forces that difference to vanish. Thus the interpolating polynomial exists and is unique.