Strong Separation of a Closed Convex Set and a Disjoint Compact Convex Set in $\mathbb{R}^n$

In the setting of Separation in $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be nonempty, closed and convex, let $K\subseteq\mathbb{R}^{n}$ be nonempty, compact and convex, and let $C\cap K=\emptyset$.

(i) Strong separation. There are $a\in\mathbb{R}^{n}$ with $a\neq0$, $\alpha\in\mathbb{R}$ and a real number $\varepsilon>0$ such that

$$a\cdot x\le\alpha-\varepsilon\quad\text{for every }x\in C\qquad\text{and}\qquad a\cdot y\ge\alpha+\varepsilon\quad\text{for every }y\in K,$$

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