The problem statement, all variables and given/known data Prove the any bounded open subset of R is the union of disjoint open intervals. The attempt at a solution I've seen a proof of this using equivalence classes, which is fine, but I want an unsophisticated solution, e.g. one relying on just the definitions of "bounded", "open" and some properties of the reals. I have an approache in mind, but I don't think it works: Let S be a bounded open subset of R. Then S is contained in contained in some open interval (-M, M). If we remove a point x from (-M, M), we get a disjoint union of open intervals, namley (-M, x) union (x, M). If we remove another point, we still get a disjoint union of open intervals. Thus, if we remove all the points of (-M, M) - S from (-M, M), which leaves S, we obtain S as the disjoint union of open intervals. The problem with this approach is that I'm removing an uncountably infinite amount of points one by one, which seems like a dubious process to me.