Let $C[0,1]$ be the space of real continuous functions on the interval $[0,1]$. A mapping $L: C[0,1] \rightarrow C[0,1]$ is said to be positive if $L(f) \geqslant 0$ for each $f \in C[0,1]$ with $f \geqslant 0$, and linear if $L(a f+b g)=a L(f)+b L(g)$ for all functions $f, g \in C[0,1]$ and constants $a, b \in \mathbb{R}$.

(i) Let $L_{n}: C[0,1] \rightarrow C[0,1]$ be a sequence of positive, linear mappings such that $L_{n}(f) \rightarrow f$ uniformly on $[0,1]$ for the three functions $f(x)=1, x, x^{2}$. Prove that $L_{n}(f) \rightarrow f$ uniformly on $[0,1]$ for every $f \in C[0,1]$.

(ii) Define $B_{n}: C[0,1] \rightarrow C[0,1]$ by

$$B_{n}(f)(x)=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}$$

where $\left(\begin{array}{l}n \\ k\end{array}\right)=\frac{n !}{k !(n-k) !}$. Using the result of part (i), or otherwise, prove that $B_{n}(f) \rightarrow f$ uniformly on $[0,1]$.

(iii) Let $f \in C[0,1]$ and suppose that

$$\int_{0}^{1} f(x) x^{4 n} d x=0$$

for each $n=0,1, \ldots$ Prove that $f$ must be the zero function.

[You should not use the Stone-Weierstrass theorem without proof.]