(a) For a positive weight function $w$, let

$$\int_{-1}^{1} f(x) w(x) d x \approx \sum_{i=0}^{n} a_{i} f\left(x_{i}\right)$$

be the corresponding Gaussian quadrature with $n+1$ nodes. Prove that all the coefficients $a_{i}$ are positive.

(b) The integral

$$I(f)=\int_{-1}^{1} f(x) w(x) d x$$

is approximated by a quadrature

$$I_{n}(f)=\sum_{i=0}^{n} a_{i}^{(n)} f\left(x_{i}^{(n)}\right)$$

which is exact on polynomials of degree $\leqslant n$ and has positive coefficients $a_{i}^{(n)}$. Prove that, for any $f$ continuous on $[-1,1]$, the quadrature converges to the integral, i.e.,

$$\left|I(f)-I_{n}(f)\right| \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty$$

[You may use the Weierstrass theorem: for any $f$ continuous on $[-1,1]$, and for any $\epsilon>0$, there exists a polynomial $Q$ of degree $n=n(\epsilon, f)$ such that $\left.\max _{x \in[-1,1]}|f(x)-Q(x)|<\epsilon .\right]$