The Convex Hull of a Compact Subset of $\mathbb{R}^n$ is Compact

In the setting of Convex Sets in $\mathbb{R}^n$: Standing Notation, with the convex hull $\operatorname{conv}$.

(i) Compact sets. If $S\subseteq\mathbb{R}^{n}$ is compact, then $\operatorname{conv}S$ is compact.

(ii) Finitely many points. For $N\in\mathbb{N}$ and $x_{1},\dots,x_{N}\in\mathbb{R}^{n}$, the set $\operatorname{conv}\{x_{1},\dots,x_{N}\}$ is compact.