Suppose $G \subset (0, \infty) $ and is not bounded. If we define set $D$ as below

$$ D = \Big\{x \in (0, \infty) : \exists A \subset \mathbb{N} \text{ not bounded s.t. } \forall n \in A, nx \in G \Big\} $$

Prove that $D$ is dense in $(0,\infty)$.


closed as unclear what you're asking by Bungo, Demophilus, TomGrubb, Paramanand Singh, Claude Leibovici Nov 16 '17 at 8:08

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  • $\begingroup$ n is a natural number? $\endgroup$ – polbos Nov 16 '17 at 0:36
  • $\begingroup$ @polbos Yes, n is a natural number $\endgroup$ – FreeMind Nov 16 '17 at 17:31

First show that $ D = (0, \infty) $.

Then show that a set with no isolated points is dense in itself.

Hence $D$ is dense in $(0, \infty)$.

  • $\begingroup$ I have edited the question. $\endgroup$ – FreeMind Nov 17 '17 at 17:17
  • $\begingroup$ @FreeMind I have edited the answer accordingly and also suggested a rewording of the question to make it clearer. $\endgroup$ – mucciolo Nov 17 '17 at 19:58

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