I'm trying to solve following question from D Kincaid & W Cheney, Numerical Analysis section 6.2 and need a proof check
Prove that if $p$ is a polynomial of degree at most $n$ then
$$ p(x) = \sum_{i=0}^{n} p[x_0,\dots,x_i] \prod_{j=0}^{i-1}(x-x_j). $$
My attempt to solve this is using the uniqueness of interpolation polynomial of degree $n$, since we have a polynomial with degree at most $n$ and we are using $n+1$ data points to interpolate that polynomial we get a polynomial with degree at most $n$ so now we have two polynomials with degree at most $n$ and they both interpolate same points so they most be the same.