Prove that if A = R in (R; Euclidean metric) then the set A must be infinite.
Where A is a closed set.
Surely A is Infinite because it's equal to an infinite set.
How will you prove this ???
Even after the correction in blue above, the question still makes no sense. Are you doing a translation of the original source?
The bit in red above worries me most. If right, it seems to trivialize the question.
As for your correction, the notation $\displaystyle A^{\prime}$ as several different meanings in mathematics. In topology it is widely used to denote the set of all limit points of $\displaystyle A$, 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 $\displaystyle A^{\prime}$. 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.