Numerical Analysis-Proof That Sum of Lagrange Interpolating Polynomials is One

Question

I am having trouble proving that given interpolation nodes $x_0, x_1, \ldots, x_n$ and function values $f_0, f_1, \ldots, f_n$ that $\sum\limits_{i=0}^n l_i(x) = 1$.

Our Lagrange interpolation formula is given by: $p_n(x) = \sum\limits_{i=0}^n f(x_i) l_i(x) $

I was able to find a proof online here: http://sepwww.stanford.edu/sep/sergey/128A/answers4.pdf It is listed as problem number 1. However, this proof does not make sense to me because it assumes that we may set $f_0, \ldots ,f_n = 1$. I would like to know how to prove that $\sum\limits_{i=0}^n l_i(x) = 1$ when we are given arbitrary $f_0, \ldots ,f_n$.

Answer 1

Note that the basis polynomials $l_i(x)$ depend only on the nodes and are therefore the same for any function values.

Also, the n-degree interpolating polynomial through n+1 points is unique, this is just "the Lagrange form" of that unique polynomial.

Apply the Lagrange interpolation formula to the polynomial $p(x)=1$