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

We first dispose of the empty case of (i): if $S=\emptyset$, then $\emptyset$ is a convex subset of $\mathbb{R}^{n}$ containing $S$ (Operations Preserving Convexity in $\mathbb{R}^n$: Intersections, Translates, Scalar Multiples, Sums, Affine Images and Preimages, Closure and Interior (i)), so $\operatorname{conv}S=\emptyset$ by Convex Hull of a Subset of $\mathbb{R}^n$ (i), and $\emptyset$ is compact. Throughout, let $S\subseteq\mathbb{R}^{n}$ be compact and nonempty. For $A,B\subseteq\mathbb{R}^{n}$ we write $J(A,B)=\{(1-s)a+sb:a\in A,\ b\in B,\ s\in[0,1]\}$ for the join of The Join of Two Compact Subsets of $\mathbb{R}^n$ is Compact (i), which is compact when $A$ and $B$ are.

Combinations of $m$ points. For $m\in\mathbb{N}$ let $C_{m}$ be the set of convex combinations of $m$ points of $S$ (Convex Weights, the Standard Simplex, and Convex Combinations in $\mathbb{R}^n$ (iii)). We first note three facts. First, $C_{1}=S$, by Convex Combinations: Small Cases, Splitting Off the Last Point, Dropping a Zero Weight, Joining Two Combinations, and Membership in Convex Sets (i). Second, $C_{m}\subseteq C_{m+1}$: given $z\in C_{m}$, pick any $v\in S$; Convex Combinations: Small Cases, Splitting Off the Last Point, Dropping a Zero Weight, Joining Two Combinations, and Membership in Convex Sets (v), applied with $P=1$, $x'_{1}=v$, $\sigma=(1)\in\Delta_{1}$ (Convex Combinations: Small Cases, Splitting Off the Last Point, Dropping a Zero Weight, Joining Two Combinations, and Membership in Convex Sets (i)) and $\lambda=0$, writes $z=(1-0)z+0\cdot v$ as a convex combination of $m+1$ points of $S$. By induction on $m$, every $C_{m}$ is nonempty; by induction on $n+1-N$, $C_{N}\subseteq C_{n+1}$ for every $N\in\mathbb{N}$ with $N\le n+1$. Third,

$$C_{m+1}=J(C_{m},S).$$

For the inclusion $\subseteq$, let $z=\sum_{k=1}^{m+1}t_{k}x_{k}$ with $x_{k}\in S$ and $t\in\Delta_{m+1}$. If $t_{m+1}<1$, Convex Combinations: Small Cases, Splitting Off the Last Point, Dropping a Zero Weight, Joining Two Combinations, and Membership in Convex Sets (iii) gives the point $y=\sum_{k=1}^{m}\tau_{k}x_{k}\in C_{m}$, with the weights $\tau\in\Delta_{m}$ it names, such that $z=(1-t_{m+1})y+t_{m+1}x_{m+1}$, and $0\le t_{m+1}\le1$; if $t_{m+1}=1$, the same clause gives $z=x_{m+1}=(1-1)y+1\cdot x_{m+1}$ for any $y\in C_{m}$, which is nonempty. In both cases $z\in J(C_{m},S)$. For the inclusion $\supseteq$, a point $(1-s)y+sv$ with $y\in C_{m}$, $v\in S$ and $s\in[0,1]$ is a convex combination of $m+1$ points of $S$ by Convex Combinations: Small Cases, Splitting Off the Last Point, Dropping a Zero Weight, Joining Two Combinations, and Membership in Convex Sets (v) with $P=1$, $x'_{1}=v$ and $\lambda=s$, together with Convex Combinations: Small Cases, Splitting Off the Last Point, Dropping a Zero Weight, Joining Two Combinations, and Membership in Convex Sets (i). Now, by induction on $m$, $C_{1}=S$ is compact, and if $C_{m}$ is compact then so is $C_{m+1}=J(C_{m},S)$ by The Join of Two Compact Subsets of $\mathbb{R}^n$ is Compact (i); so every $C_{m}$ is compact.

Proof of (i). The empty case was treated above; for the nonempty compact $S$ we show $\operatorname{conv}S=C_{n+1}$. The inclusion $C_{n+1}\subseteq\operatorname{conv}S$ holds by The Convex Hull Consists of the Convex Combinations of Finitely Many Points (i). Conversely, for $z\in\operatorname{conv}S$, Carathéodory's Theorem in $\mathbb{R}^n$ (i) gives $N\in\mathbb{N}$ with $N\le n+1$ and $z\in C_{N}$, and $C_{N}\subseteq C_{n+1}$ by the second fact above. Thus $\operatorname{conv}S=C_{n+1}$, which is compact.

Proof of (ii). Let $F=\{x_{k}:k\in[N]\}$. Then $F$ is compact. Indeed, let $\mathcal{U}$ be a family of open sets of the subspace topology of $F$ covering $F$ (Compact Topological Space and Compact Subset (i), Compact Topological Space and Compact Subset (ii)). For each $k\in[N]$ the set $\{U\in\mathcal{U}:x_{k}\in U\}$ is nonempty, so, the index set being finite, Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (xi) gives sets $U_{k}\in\mathcal{U}$ with $x_{k}\in U_{k}$ for every $k\in[N]$. The image $\{U_{k}:k\in[N]\}$ is a finite subfamily of $\mathcal{U}$ (Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii), Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vi)) and covers $F$. By (i), $\operatorname{conv}F$ is compact.