Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous and let $n$ be a positive integer. For $g:[0,1] \rightarrow \mathbb{R}$ a continuous function, write $\|f-g\|_{L^{\infty}}=\sup _{x \in[0,1]}|f(x)-g(x)|$.

(i) Let $p$ be a polynomial of degree at most $n$ with the property that there are $(n+2)$ distinct points $x_{1}, x_{2}, \ldots, x_{n+2} \in[0,1]$ with $x_{1}<x_{2}<\ldots<x_{n+2}$ such that

$$f\left(x_{j}\right)-p\left(x_{j}\right)=(-1)^{j}\|f-p\|_{L^{\infty}}$$

for each $j=1,2, \ldots, n+2$. Prove that

$$\|f-p\|_{L^{\infty}} \leqslant\|f-q\|_{L^{\infty}}$$

for every polynomial $q$ of degree at most $n$.

(ii) Prove that there exists a polynomial $p$ of degree at most $n$ such that

$$\|f-p\|_{L^{\infty}} \leqslant\|f-q\|_{L^{\infty}}$$

for every polynomial $q$ of degree at most $n$.

[If you deduce this from a more general result about abstract normed spaces, you must prove that result.]

(iii) Let $Y=\left\{y_{1}, y_{2}, \ldots, y_{n+2}\right\}$ be any set of $(n+2)$ distinct points in $[0,1]$.

(a) For $j=1,2, \ldots, n+2$, let

$$r_{j}(x)=\prod_{k=1, k \neq j}^{n+2} \frac{x-y_{k}}{y_{j}-y_{k}}$$

$t(x)=\sum_{j=1}^{n+2} f\left(y_{j}\right) r_{j}(x)$ and $r(x)=\sum_{j=1}^{n+2}(-1)^{j} r_{j}(x)$. Explain why there is a unique number $\lambda \in \mathbb{R}$ such that the degree of the polynomial $t-\lambda r$ is at $\operatorname{most} n$.

(b) Let $\|f-g\|_{L^{\infty}(Y)}=\sup _{x \in Y}|f(x)-g(x)|$. Deduce from part (a) that there exists a polynomial $p$ of degree at most $n$ such that

$$\|f-p\|_{L^{\infty}(Y)} \leqslant\|f-q\|_{L^{\infty}(Y)}$$

for every polynomial $q$ of degree at most $n$.