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

In the setting of Separation in $\mathbb{R}^n$: Standing Notation, let $C,D\subseteq\mathbb{R}^{n}$ be nonempty, convex and disjoint.

(i) Separation. There are $a\in\mathbb{R}^{n}$ with $a\neq0$ and $\alpha\in\mathbb{R}$ such that

$$a\cdot x\le\alpha\le a\cdot y\qquad\text{for all }x\in C\text{ and }y\in D,$$

so that the hyperplane $H(a,\alpha)$ separates $C$ and $D$.