State the Stone-Weierstrass Theorem for real-valued functions.

State Riesz's Lemma.

Let $K$ be a compact, Hausdorff space and let $A$ be a subalgebra of $C(K)$ separating the points of $K$ and containing the constant functions. Fix two disjoint, non-empty, closed subsets $E$ and $F$ of $K$.

(i) If $x \in E$ show that there exists $g \in A$ such that $g(x)=0,0 \leqslant g<1$ on $K$, and $g>0$ on $F$. Explain briefly why there is $M \in \mathbb{N}$ such that $g \geqslant \frac{2}{M}$ on $F$.

(ii) Show that there is an open cover $U_{1}, U_{2}, \ldots, U_{m}$ of $E$, elements $g_{1}, g_{2}, \ldots, g_{m}$ of $A$ and positive integers $M_{1}, M_{2}, \ldots, M_{m}$ such that

$$0 \leqslant g_{r}<1 \text { on } K, \quad g_{r} \geqslant \frac{2}{M_{r}} \text { on } F, \quad g_{r}<\frac{1}{2 M_{r}} \text { on } U_{r}$$

for each $r=1,2, \ldots, m$.

(iii) Using the inequality

$$1-N t \leqslant(1-t)^{N} \leqslant \frac{1}{N t} \quad(0<t<1, N \in \mathbb{N})$$

show that for sufficiently large positive integers $n_{1}, n_{2}, \ldots, n_{m}$, the element

$$h_{r}=1-\left(1-g_{r}^{n_{r}}\right)^{M_{r}^{n_{r}}}$$

of $A$ satisfies

$$0 \leqslant h_{r} \leqslant 1 \text { on } K, \quad h_{r} \leqslant \frac{1}{4} \text { on } U_{r}, \quad h_{r} \geqslant\left(\frac{3}{4}\right)^{\frac{1}{m}} \text { on } F$$

for each $r=1,2, \ldots, m$.

(iv) Show that the element $h=h_{1} \cdot h_{2} \cdots \cdot h_{m}-\frac{1}{2}$ of $A$ satisfies

$$-\frac{1}{2} \leqslant h \leqslant \frac{1}{2} \text { on } K, \quad h \leqslant-\frac{1}{4} \text { on } E, \quad h \geqslant \frac{1}{4} \text { on } F \text {. }$$

Now let $f \in C(K)$ with $\|f\| \leqslant 1$. By considering the sets $\left\{x \in K: f(x) \leqslant-\frac{1}{4}\right\}$ and $\left\{x \in K: f(x) \geqslant \frac{1}{4}\right\}$, show that there exists $h \in A$ such that $\|f-h\| \leqslant \frac{3}{4}$. Deduce that $A$ is dense in $C(K)$.