Proof of A Point Outside the Interior of a Closed Convex Subset of $\mathbb{R}^n$ Has a Supporting Unit Normal

Throughout, closedness and compactness in $\mathbb{R}^{n}$ refer to the metric topology $\mathcal{T}_{d_{E}}$, which exists by The Open Sets of a Metric Space Form a Set, Given by a Predicative Formula (ii). We find, for each $\delta>0$, a point of $\mathbb{S}$ that supports $C$ at $x$ up to an error $\delta$, and then remove the error by compactness of $\mathbb{S}$.

For a real number $\delta>0$ let

$$F_{\delta}=\{a\in\mathbb{R}^{n}:\ a\cdot z\le a\cdot x+\delta\text{ for every }z\in C\}.$$

With $W=\{z-x:z\in C\}$, symmetry and bilinearity of the dot product give $a\cdot z-a\cdot x=(z-x)\cdot a$, so

$$F_{\delta}=\{a\in\mathbb{R}^{n}:\ w\cdot a\le\delta\text{ for every }w\in W\},$$

and $F_{\delta}$ is closed by Sets Cut Out by Linear Inequalities in $\mathbb{R}^n$ Are Closed (ii). Clearly $F_{\delta}\subseteq F_{\delta'}$ whenever $0<\delta\le\delta'$.

We claim that $\mathbb{S}\cap F_{\delta}\neq\emptyset$ for every $\delta>0$. Fix $\delta>0$. Since $x\notin\operatorname{int}C$, the point $x$ is not an interior point of $C$ (Interior of a Subset of a Metric Space (ii)), so the open ball $B(x,\delta)$ is not contained in $C$: there is $y\in\mathbb{R}^{n}\setminus C$ with $\lVert y-x\rVert<\delta$. By Strong Separation of a Point from a Closed Convex Subset of $\mathbb{R}^n$ (i), applied to $C$ and $y$, there is $b\in\mathbb{R}^{n}$ with $b\neq0$ and $b\cdot z<b\cdot y$ for every $z\in C$. The point $a=b/\lVert b\rVert\in\mathbb{S}$ then satisfies $a\cdot z<a\cdot y$ for $z\in C$, and by The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ (i)

$$a\cdot z<a\cdot y=a\cdot x+a\cdot(y-x)\le a\cdot x+\lVert y-x\rVert<a\cdot x+\delta\qquad(z\in C).$$

So $a\in\mathbb{S}\cap F_{\delta}$, proving the claim.

Now we treat all $\delta$ at once. Apply The Finite Intersection Property of a Compact Set (i), in $(\mathbb{R}^{n},\mathcal{T}_{d_{E}})$, to the compact set $\mathbb{S}$, the nonempty index set $I=\{\delta\in\mathbb{R}:\delta>0\}$ and the family $(F_{\delta})_{\delta\in I}$ of closed sets. Its hypothesis holds: a nonempty finite $J\subseteq I$ has a least element $\delta_{0}$ (Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (viii)), and by the monotonicity of the sets $F_{\delta}$ and the claim,

$$\emptyset\neq\mathbb{S}\cap F_{\delta_{0}}\subseteq\mathbb{S}\cap\bigcap_{\delta\in J}F_{\delta}.$$

Hence there is $a\in\mathbb{S}$ lying in $F_{\delta}$ for every $\delta>0$.

Let $z\in C$. Then $a\cdot z\le a\cdot x+\delta$ for every $\delta>0$. If we had $a\cdot z>a\cdot x$, the choice $\delta=\tfrac{1}{2}(a\cdot z-a\cdot x)>0$ would violate this. So $a\cdot z\le a\cdot x$ for every $z\in C$, which is (i).