Let $K$ be a non-empty compact Hausdorff space and let $C(K)$ be the space of real-valued continuous functions on $K$.

(i) State the real version of the Stone-Weierstrass theorem.

(ii) Let $A$ be a closed subalgebra of $C(K)$. Prove that $f \in A$ and $g \in A$ implies that $m \in A$ where the function $m: K \rightarrow \mathbb{R}$ is defined by $m(x)=\max \{f(x), g(x)\}$. [You may use without proof that $f \in A$ implies $|f| \in A$.]

(iii) Prove that $K$ is normal and state Urysohn's Lemma.

(iv) For any $x \in K$, define $\delta_{x} \in C(K)^{*}$ by $\delta_{x}(f)=f(x)$ for $f \in C(K)$. Justifying your answer carefully, find

$$\inf _{x \neq y}\left\|\delta_{x}-\delta_{y}\right\| .$$