 Solution

For distinct nodes, define the divided difference recursively byOnly the function values at the nodes are needed. Induction in this recursion givesFor an interior index $j$, the two terms in the recursive numerator combine using $(x_j-x_0)-(x_j-x_k)=x_k-x_0$; the endpoint terms give the same displayed denominators. This proves the formula, including its symmetry in the nodes.

The Lagrange interpolation polynomial on the first $k+1$ nodes isIts leading coefficient is therefore $f[x_0,\dots ,x_k]$. Meanwhile $p_k-p_{k-1}$ vanishes at $x_0,\dots ,x_{k-1}$ and has degree at most $k$. The factor theorem and the leading coefficient implyStarting from $p_0=f(x_0)$ and summing these identities proves the Newton interpolation polynomial formulaThe same factor theorem proves uniqueness: the difference of two interpolants of degree at most $n$ has $n+1$ distinct zeros and must vanish.