It is well known that disjoint closed sets in $\mathbb{R}^n$ can have distance equal to 0 (take, for example, the curves $y=\pm\frac{1}{x}$). If we add the requirement that the sets be bounded, this is no longer possible, since such sets are compact by the Heine-Borel theorem.
My question is whether there exists an elementary proof of this fact. I realize that I could surreptitiously inline the relevant parts of the proof of Heine-Borel, but that's not very interesting.
While thinking about this, I came to believe that it should be slightly easier to prove this for convex sets. The key here seems to be the proposition that two closed bounded convex sets can be separated by open balls. I'm not sure how much of a simplification this is as I don't have much background in geometry; the only relevant result that comes to mind is the separation of convex sets in Banach spaces, and this looks to be way overkill. I'd appreciate any input on this as well.