Separation via Sup/Inf Inequality

Let $X$ be a real vector space and let $\Omega_1,\Omega_2\subset X$ be nonempty.

Proposition: The sets $\Omega_1$ and $\Omega_2$ can be separated by a hyperplane if and only if there exists a nonzero linear functional $f:X\to\mathbb{R}$ such that

Context: This equivalence is often used because it replaces a pointwise inequality (“for all $x$ and $y$”) with a single inequality between two real numbers. In the notes, the existence of $\sup$ and $\inf$ is justified using completeness of $\mathbb{R}$.