(i) Suppose that $f:[0,1] \rightarrow \mathbb{R}$ is continuous. Prove the theorem of Bernstein which states that, if we write

$$f_{m}(t)=\sum_{r=0}^{m}\left(\begin{array}{c} m \\ r \end{array}\right) f(r / m) t^{r}(1-t)^{m-r}$$

for $0 \leqslant t \leqslant 1$, then $f_{m} \rightarrow f$ uniformly as $m \rightarrow \infty$

(ii) Let $n \geqslant 1, a_{1, n}, a_{2, n}, \ldots, a_{n, n} \in \mathbb{R}$ and let $x_{1, n}, x_{2, n}, \ldots, x_{n, n}$ be distinct points in $[0,1]$. We write

$$I_{n}(g)=\sum_{j=1}^{n} a_{j, n} g\left(x_{j, n}\right)$$

for every continuous function $g:[0,1] \rightarrow \mathbb{R}$. Show that, if

$$I_{n}(P)=\int_{0}^{1} P(t) d t$$

for all polynomials $P$ of degree $2 n-1$ or less, then $a_{j, n} \geqslant 0$ for all $1 \leqslant j \leqslant n$ and $\sum_{j=1}^{n} a_{j, n}=1 .$

(iii) If $I_{n}$ satisfies the conditions set out in (ii), show that

$$I_{n}(f) \rightarrow \int_{0}^{1} f(t) d t$$

as $n \rightarrow \infty$ whenever $f:[0,1] \rightarrow \mathbb{R}$ is continuous.