For $N\in\mathbb{N}$, let:
$S_{N}=\left\{ \left(\frac{2}{9}\right)^{N},\left(\frac{2}{9}\right)^{N}\times6,\left(\frac{2}{9}\right)^{N}\times6^{2},...,\left(\frac{2}{9}\right)^{N}\times6^{2N}\right\}$
I suspect that, as $N\rightarrow\infty$, $S_{N}$ tends to a dense subset of $[0,\infty)$. How would one go about (dis)proving this?
Addendum:
Consider a sequence $\left\{ s_{N}\right\} _{N\in\mathbb{N}}$ such that for each $N\in\mathbb{N}$, $s_{N}\in S_{N}$. Do there exist such sequences which converge to 1 as $N\rightarrow\infty$? And what can be said about the set of the limits of all such sequences?
The gap in the sequence is about the same as the last element, so none of the $S_N$ are dense away from a neighborhood of zero. Their union, however does become dense.