Wow. You have no idea what a Dedekind cut is or what it's for.
First, a Dedekind cut is an equivalence class of pairs of rationals, with the usual lower and upper properties. The fact that you started with L and R being sets of reals immediately shows that they are NOT Dedekind cuts. This isn't a typo or trivial point. It's the entire essence of the concept. Dedekind cuts are a way of constructing the reals given just the rationals. If you started with the reals you wouldn't need Dedekind cuts. So this error shows that you are completely unclear on the concept.
The idea for example is that the irrational number $$\sqrt 2$$ is the equivalence class of the pair of lower and upper sets: $$\}q \in \mathbb Q : q^2 < 2\}$$ and $$\{q \in \mathbb Q : q^2 > 2\}$$. You can see that the lower set has no least upper bound in the rationals; and the upper set has no greatest lower bound in the rationals. So we DEFINE $$\sqrt 2$$ to be that cut.
But you see Dedekind cuts are used to DEFINE the reals, so it makes no sense to start by assuming you know what the reals are.
What you are trying to say is that the real numbers have the least upper bound property. Every nonempty set that's bounded above has a least upper bound. This condition is equivalent to completeness, the statement that every Cauchy sequence converges. It means that the set in question, in this case the reals, has no "holes." That's what it means. If someone doesn't like the word holes then use the word foozles. The concept is the same and it's the defining attribute of the standard real numbers.
If you explained completeness in terms people could understand, that would be useful. But to invoke Dedekind cuts is completely wrong in this context, even if you had the math right, which you totally didn't. Because Dedekind cuts are a difficult construction, shown to people once so they know that there's a set-theoretic construction of the reals. Once presented, nobody ever uses it again. It's entirely inappropriate in the current context of simply trying to say why the least upper bound property is important for the reals.
It's wildly off the mark for an exposition at the level you were intending, even if you got the math right by starting with the rationals. But to completely misunderstand the subject and then try to defend your misunderstandings shows, it's fair to say, a certain lack of self-awareness about the limitations of one's grasp of the subject of real analysis; what Dedekind cuts are; and what their purpose is in the scheme of things.
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