Sets Cut Out by Linear Inequalities in $\mathbb{R}^n$ Are Closed

In the setting of Separation in $\mathbb{R}^n$: Standing Notation.

(i) One inequality. For every $w\in\mathbb{R}^{n}$ and $\beta\in\mathbb{R}$, the set $\{y\in\mathbb{R}^{n}:w\cdot y\le\beta\}$ is closed.

(ii) Many inequalities. For every $W\subseteq\mathbb{R}^{n}$ and $\beta\in\mathbb{R}$, the set

$$\{y\in\mathbb{R}^{n}:\ w\cdot y\le\beta\text{ for every }w\in W\}$$

is closed.

(iii) Hyperplanes and closed half-spaces. For every $a\in\mathbb{R}^{n}$ with $a\neq0$ and every $\alpha\in\mathbb{R}$, the hyperplane $H(a,\alpha)$ and the closed half-spaces $H^{\le}(a,\alpha)$ and $H^{\ge}(a,\alpha)$ are closed.