Proof of Existence and Uniqueness of the Smallest Convex Superset of a Subset of $\mathbb{R}^n$

No choice is used; every object below is defined by a formula. Let $\mathcal{C}$ be the set of all $C\in\mathcal{P}(\mathbb{R}^{n})$ that are convex and contain $S$.

Let

$$H=\bigcap\mathcal{C},$$

which is exactly the set displayed in (ii). Then $H$ is convex by Operations Preserving Convexity in $\mathbb{R}^n$: Intersections, Translates, Scalar Multiples, Sums, Affine Images and Preimages, Closure and Interior (ii); it contains $S$, because every member of $\mathcal{C}$ does; and it is contained in every convex $C\supseteq S$, because such a $C$ is a member of $\mathcal{C}$. So $H$ has the property required in (i).

For uniqueness, suppose that $H$ and $H'$ both have this property. Each is a convex superset of $S$, so the property of $H$ gives $H\subseteq H'$ and the property of $H'$ gives $H'\subseteq H$; hence $H=H'$. This proves (i), and since the unique such set is the intersection $H$ constructed above, it also proves (ii).