So far in my class, we've defined the real numbers as a collection of all Dedekind cuts. Additionally, we've proven that any nonempty subset of the real numbers that is bounded above has a supremum. Using this, I am asked to prove that the real numbers are connected (i.e., they can't be assembled from the union of two nonempty, disjoint, and open sets). We have already proven that a continuum (we have not yet defined the real numbers as the only continuum) is connected if all bounded and nonempty subsets have supremum. However, the proof is very long and complicated and has nothing to do with Dedekind cuts; while I could just recycle this old proof, I'm sure there is probably a simpler proof I am missing that can use the fact that the real numbers are made of Dedekind cuts. Any proofs or hints would be greatly appreciated.
By the way, if anything I've said does not make sense or if you'd like me to provide more information about anything, just let me know.