Lagrange interpolation of a polynomial

Question

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ has such property that for every distinct $x_0,x_1,...,x_n\in\mathbb{R}$ Lagrange interpolating polynomial for $f$ in these points has degree at most $n-1$. Prove that $f$ is a polynomial.

Completely don't know how to start.

Answer 1

To condense my comments: Consider the Lagrange interpolation polynomials

• $q$ for the $n$ points $x_1,x_2,\dots,x_n$ and
• $p_x$ for the $n+1$ points $x,x_1,\dots,x_n$,

under the assumption that the points $x,x_1,x_2,\dots,x_n$ are all different. Then what difference is there between the coefficients of $q$ and $p_x$ and what does that tell about the values of $p_x$, $q$, and $f$ at the point $x$?