proof of uniqueness of Lagrange Interpolation formula

Existence is clear from the construction, the uniqueness is proved by assuming there are two different polynomials $p(x)$ and $q(x)$ that interpolate the points. Then $r(x)=p(x)-q(x)$ has $n$ zeros, $x_1,\mathrm{\dots },x_n$ and there is a point $x_e$ such that $r(x_e)\ne 0$. $r(x)$ is non-constant with degree $\mathrm{deg}(r(x))\le n-1$ and has more than $n-1$ solutions, which is a contradiction. Thus there can only be one polynomial.

Title	proof of uniqueness of Lagrange Interpolation formula
Canonical name	ProofOfUniquenessOfLagrangeInterpolationFormula
Date of creation	2013-03-22 14:09:25
Last modified on	2013-03-22 14:09:25
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	10
Author	rspuzio (6075)
Entry type	Proof
Classification	msc 65D05
Classification	msc 41A05