I have read the following proof that two disjoint compact sets in Euclidean space can be separated:
Assume $C,D$ are nonempty.
Let $C,D\subset \Bbb R^d$ be disjoint convex compact sets. Then the cartesian product $C\times D$ is a compact space, too, and the distance function $(x,y)\mapsto \|x-y\|$ attains its minimum on $C\times D$. That is, there exist points $p\in C$ and $q\in D$ such that the distance of $C$ and $D$ equals the distance of $p$ and $q$. The desired separating hyperplane can be taken to be the one perpendicular to the segment $pq$ and passing through its midpoint.
I find this proof very strangely written. 1) If we are defining a function from $C\times D\to \Bbb R$, then of course it attains its minimum on its domain, that's tautological? 2) I don't see how compactness of $C\times D$ was used.
The compactness was used to guarantee, via Weierstrass, that the distance function attains its minimum on $C \times D$.
It is not true that functions always attain their minimum on their domain; one way to guarantee that they do is via compactness (and continuity).
For example, consider the function $f(x)=x$ defined on $(0,1)$. Clearly, $f$ does not attain its minimum in its domain.