(a) State the real version of the Stone-Weierstrass theorem and state the UrysohnTietze extension theorem.

(b) In this part, you may assume that there is a sequence of polynomials $P_{i}$ such that $\sup _{x \in[0,1]}\left|P_{i}(x)-\sqrt{x}\right| \rightarrow 0$ as $i \rightarrow \infty$.

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous piecewise linear function which is linear on $[0,1 / 2]$ and on $[1 / 2,1]$. Using the polynomials $P_{i}$ mentioned above (but not assuming any form of the Stone-Weierstrass theorem), prove that there are polynomials $Q_{i}$ such that $\sup _{x \in[0,1]}\left|Q_{i}(x)-f(x)\right| \rightarrow 0$ as $i \rightarrow \infty$.

(d) Which of the following families of functions are relatively compact in $C[0,1]$ with the supremum norm? Justify your answer.

$$\begin{aligned} &\mathcal{F}_{1}=\left\{x \mapsto \frac{\sin (\pi n x)}{n}: n \in \mathbb{N}\right\} \\ &\mathcal{F}_{2}=\left\{x \mapsto \frac{\sin (\pi n x)}{n^{1 / 2}}: n \in \mathbb{N}\right\} \\ &\mathcal{F}_{3}=\{x \mapsto \sin (\pi n x): n \in \mathbb{N}\} \end{aligned}$$

[In this question $\mathbb{N}$ denotes the set of positive integers.]