Proof of A Closed Subset of a Euclidean Space is a Countable Union of Compact Sets, and Its Coordinate Projections are Borel

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here.

Step 1 (Claim 1). Let $C\subseteq\mathbb{R}^{m}$ be closed and let $C_{k}=C\cap\bar{B}(0_{\mathbb{R}^{m}},\iota(k))$ for $k\in\mathbb{N}$. Each $C_{k}$ is compact. Indeed $\iota(k)$ is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so the closed ball $\bar{B}(0_{\mathbb{R}^{m}},\iota(k))$ is closed by claim 3 of Elementary Properties of the Closed Ball in a Metric Space, and $C_{k}$, the intersection of the family consisting of $C$ and this ball, is closed by claim 3 of Complements, Unions and Intersections of Closed Sets in a Topological Space. Every $y\in C_{k}$ satisfies $d_{E}(0_{\mathbb{R}^{m}},y)\le\iota(k)$ by Closed Ball in a Metric Space, so $C_{k}$ is bounded in the sense of Bounded Subset of a Metric Space, with centre $0_{\mathbb{R}^{m}}$ and $R=\iota(k)$. Hence $C_{k}$ is compact by The Heine-Borel Theorem in $\mathbb{R}^n$, and Compactness of Closed Balls and the Unit Sphere, claim 2 there implying claim 1. Every $C_{k}$ is a subset of $C$, so $\bigcup_{k\in\mathbb{N}}C_{k}\subseteq C$. Conversely let $x\in C$. The Euclidean norm $\lVert x\rVert$ is a real number, so by claim 1 of The Archimedean Property of the Real Numbers there is $k\in\mathbb{N}$ with $\lVert x\rVert<\iota(k)$. By claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square the Euclidean distance $d_{E}(x,0_{\mathbb{R}^{m}})$ equals $\lVert x-0_{\mathbb{R}^{m}}\rVert=\lVert x\rVert$, so $x\in\bar{B}(0_{\mathbb{R}^{m}},\iota(k))$ and hence $x\in C_{k}$. Therefore $C=\bigcup_{k\in\mathbb{N}}C_{k}$.

Step 2 (Claim 2). Let $C\subseteq\mathbb{R}^{q+p}$ be closed and let $C_{k}$ be as in claim 1, applied with $m=q+p$. The projection $\mathrm{pr}^{q,p}_{1}:\mathbb{R}^{q+p}\to\mathbb{R}^{q}$ is continuous for the Euclidean distances: by claim 1 of Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit it is nonexpansive, hence Lipschitz with constant $1$, hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous.

By Continuity of a Map Between Metric Spaces via Preimages of Open Sets it is therefore a continuous map between the topological spaces $\mathbb{R}^{q+p}$ and $\mathbb{R}^{q}$ with their metric topologies, and for each $k$ its restriction to $C_{k}$ is continuous for the subspace topology on $C_{k}$ by claim 1 of Restriction of a Continuous Map, and Continuous Images of Compact Subsets. Since $C_{k}$ is compact in $\mathbb{R}^{q+p}$, that is, compact as a topological space with the subspace topology by Compact Topological Space and Compact Subset, the image $\mathrm{pr}^{q,p}_{1}(C_{k})$ of this restriction is compact by Continuous Image of a Compact Space is Compact, and therefore belongs to $\mathcal{B}(\mathbb{R}^{q})$ by Compact Subsets of a Metric Space are Closed and Borel (ii). Since the image of a union is the union of the images,

$$\mathrm{pr}^{q,p}_{1}(C)=\bigcup_{k\in\mathbb{N}}\mathrm{pr}^{q,p}_{1}(C_{k}),$$

which belongs to $\mathcal{B}(\mathbb{R}^{q})$, a $\sigma$-algebra being closed under countable unions.

The same argument applies to $\mathrm{pr}^{q,p}_{2}$, which is nonexpansive by claim 1 of Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit as well, and gives $\mathrm{pr}^{q,p}_{2}(C)\in\mathcal{B}(\mathbb{R}^{p})$.