Prove Bernstein's theorem, which states that if $f:[0,1] \rightarrow \mathbb{R}$ is continuous and

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

then $f_{m}(t) \rightarrow f(t)$ uniformly on $[0,1]$. [Theorems from probability theory may be used without proof provided they are clearly stated.]

Deduce Weierstrass's theorem on polynomial approximation for any closed interval.

Proving any results on Chebyshev polynomials that you need, show that, if $g:[0, \pi] \rightarrow \mathbb{R}$ is continuous and $\epsilon>0$, then we can find an $N \geqslant 0$ and $a_{j} \in \mathbb{R}$, for $0 \leqslant j \leqslant N$, such that

$$\left|g(t)-\sum_{j=0}^{N} a_{j} \cos j t\right| \leqslant \epsilon$$

for all $t \in[0, \pi]$. Deduce that $\int_{0}^{\pi} g(t) \cos n t d t \rightarrow 0$ as $n \rightarrow \infty$.