Indeed, I don't see the point. Maybe the question is : every dense subset of must be infinite.
The bit in red above worries me most. If right, it seems to trivialize the question.
As for your correction, the notation as several different meanings in mathematics. In topology it is widely used to denote the set of all limit points of , the derived set.
In some set theory textbooks the notation means the complement of the set.
I think that you must post what your text material says about . How is it used?
Also, please review the exact wording of this problem.
Okay, so that is what tinyboss suggested initially: the [b]closure[b] of A, not A itself, is equal to R- A is dense in R. I think the simplest way to prove that a set dense in R must be infinite is an indirect proof: If A were finite, then there would be a largest member, m. Then look at m+1 to show that this contradicts the fact that A is dense in R.