Let $x_{1}, x_{2}, \ldots, x_{n}$ be the roots of the Legendre polynomial of degree $n$. Let $A_{1}$, $A_{2}, \ldots, A_{n}$ be chosen so that

$$\int_{-1}^{1} p(t) d t=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$$

for all polynomials $p$ of degree $n-1$ or less. Assuming any results about Legendre polynomials that you need, prove the following results:

(i) $\int_{-1}^{1} p(t) d t=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$ for all polynomials $p$ of degree $2 n-1$ or less;

(ii) $A_{j} \geqslant 0$ for all $1 \leqslant j \leqslant n$;

(iii) $\sum_{j=1}^{n} A_{j}=2$.

Now consider $Q_{n}(f)=\sum_{j=1}^{n} A_{j} f\left(x_{j}\right)$. Show that

$$Q_{n}(f) \rightarrow \int_{-1}^{1} f(t) d t$$

as $n \rightarrow \infty$ for all continuous functions $f$.