Limit points of "harmonic" numbers

Consider the standard topology on $\Bbb R$ generated by the open intervals.

Consider the class of sets $\{A_n\}_{n\geq 0}$ where $A_n\subseteq\Bbb R$ is defined by, $A_n=\left\{\sum\limits_{k=1}^{n}\frac{1}{p_k}\mid p_k\in\Bbb N~\forall~1\leq k\leq n\right\}~\forall~n\geq 1$ and $A_0=\{0\}$

Which one of the following options is correct?

1) $(A_n)^\prime=\bigcap_{i=0}^{n-1} A_i$

2) $(A_n)^\prime=\left(\bigcup_{i\textrm{ odd}}A_i\right)\cap\left(\bigcup_{i\textrm{ even}}A_i\right)$

3) $(A_n)^\prime=\bigcup_{i=0}^{n-1} A_i$

4) $(A_n)^\prime=\left(\bigcap_{i\textrm{ odd}}A_i\right)\cup\left(\bigcap_{i\textrm{ even}}A_i\right)$

---

Details and Assumptions:

• For a subset $X$ of a topological space, $X^\prime$ denotes the derived set (set of all limit points) of $X$ .

• $\cup$ and $\cap$ denote set union and intersection respectively.

• $\Bbb N$ and $\Bbb R$ denotes the set of natural numbers and real number, respectively.