# Proving that $D$ is dense in $(0,\infty)$ [closed]

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 $(0,\infty)$.

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

First show that $D = (0, \infty)$.
Hence $D$ is dense in $(0, \infty)$.