Let x_(0)lt x_(1)lt ... lt x_(N) be N+1 distinct interpolation points. Prove th sum _(i=0)^NL_(i)(x)=1 where L_(i)(x) for i=0,1,... ,N are the Lagrange basis polynomials.

To prove the statement, we can use the definition of the Lagrange basis polynomials and the properties of interpolation.The Lagrange basis polynomials are defined as follows: Now, let's consider the sum . We can rewrite this sum as: Notice that each term in the sum is a product of terms of the form , where . When we expand this product, each term will cancel out with another term in the sum, except for the terms where . Therefore, we have: Thus, we have proved that .