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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