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 $k\in\mathbb{N}$; compactness of each $C_{k}$ is recorded in the statement. 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 Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ 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.

For each $k$ the image $\mathrm{pr}^{q,p}_{1}(C_{k})$ is compact by Continuous Image of a Compact Space is Compact, $C_{k}$ being compact, and therefore belongs to $\mathcal{B}(\mathbb{R}^{q})$ by Compact Subsets of a Metric Space are Closed and Borel §borel. Since the image of a union is the union of the images,

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 $\mathrm{pr}^{q,p}_{2}(C)\in\mathcal{B}(\mathbb{R}^{p})$.