Let $L_i(x)$ Lagrange polynomials $\prod_{i\neq j}\frac{x-x_j}{x_i-x_j}$ for $n+1$ points $x_0,x_1,...,x_n$
a. prove that for all $x$ $$\sum_{i=0}^nL_i(x)=1$$
b. prove: if $n\geq 1$, for all $x$
$$\sum_{i=0}^nxL_i(x)=x$$
So I proved a by interpolating the function $f(x)=1$ and using the fact that the interpolation polynomial is unique.
for b, can we say that because the coefficients of lagrange does not depend on the $y$ values, for the same points we can multiply the interpolation polynomial we got at a to get:
$$x=x\sum_{i=0}^nL_i(x)\Rightarrow x=\sum_{i=0}^nxL_i(x)$$