The question might sound ridiculous, but I am not able to prove it with rigor. I tried proving it by the following definitions ONLY.
Open set: A set $U$ is open if for every $a$ belonging to $U$, there is some $r = r(a) > 0$ such that the ball $B_r(a)$ is contained in $U$.
Closed set: A set $A$ is closed if it contains all of its limit points.
I tried using that $A$ closed and open would render $A'$ open and closed (provable using the above definitions) but that didn't lead me anywhere.