The Supporting Hyperplane Theorem in $\mathbb{R}^n$

In the setting of Separation in $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be convex, and let $x\in\mathbb{R}^{n}$ with $x\notin\operatorname{int}C$.

(i) Unit normal. There is $a\in\mathbb{S}$ with

$$a\cdot z\le a\cdot x\qquad\text{for every }z\in C.$$

(ii) Supporting hyperplane. Let $a$ be as in (i). If $x\in C$, the hyperplane $H(a,a\cdot x)$ supports $C$ at $x$.

(iii) Supporting hyperplane of the closure. Let $a$ be as in (i). If $x\in\operatorname{cl}C$, the hyperplane $H(a,a\cdot x)$ supports $\operatorname{cl}C$ at $x$.