# Limit points of "harmonic" numbers

Calculus Level 5

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.

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