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The Lagrange cardinal polynomials areThus the Lagrange interpolation polynomial isIt has degree at most $n$ and takes the required values. Uniqueness follows because the difference of two such polynomials has $n+1$ distinct roots and degree at most $n$, so is zero. The source's phrase “degree $n$” must allow a smaller degree, for example when $f$ is constant.

Define the divided difference recursively by $f[x_i]=f_i$ andInduction on this recursion, or comparison of the leading coefficient in the Newton interpolation polynomial, givesFor completeness the induction works term by term: an interior term gets the coefficient $(1/(x_i-x_n)-1/(x_i-x_0))/(x_n-x_0)=1/((x_i-x_n)(x_i-x_0))$, multiplying the product over the other interior nodes; the two endpoint terms give the same formula. This is also the leading coefficient of $p$ directly from the cardinal formula, so $p^{(n)}=n!f[x_0,\dots ,x_n]$.

The function $f-p\in C^n[x_0,x_n]$ has $n+1$ distinct zeros. Repeated application of Rolle theorem gives $\xi \in (x_0,x_n)$ with $(f-p{)}^{(n)}(\xi )=0$ when $n\ge 1$. HenceFor $n=0$, this is simply $f[x_0]=f(x_0)$ and we take $\xi =x_0$.