(i) State carefully the theorems of Stone-Weierstrass and Arzelá-Ascoli (work with real scalars only).

(ii) Let $\mathcal{F}$ denote the family of functions on $[0,1]$ of the form

$$f(x)=\sum_{n=1}^{\infty} a_{n} \sin (n x)$$

where the $a_{n}$ are real and $\left|a_{n}\right| \leqslant 1 / n^{3}$ for all $n \in \mathbb{N}$. Prove that any sequence in $\mathcal{F}$ has a subsequence that converges uniformly on $[0,1]$.

(iii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=0$ and $f^{\prime}(0)$ exists. Show that for each $\varepsilon>0$ there exists a real polynomial $p$ having only odd powers, i.e. of the form

$$p(x)=a_{1} x+a_{3} x^{3}+\cdots+a_{2 m-1} x^{2 m-1},$$

such that $\sup _{x \in[0,1]}|f(x)-p(x)|<\varepsilon$. Show that the same holds without the assumption that $f$ is differentiable at 0 .