Construct a random walk on $\mathbb{R}$ such that it is returnable and the set of its values is everywhere dense in $\mathbb{R}$.
I think that random walk $S_{n} = \xi_{1} + ... + \xi_{n}$, where random variable $\xi$ with values $\frac{1}{n}$, $n \in \mathbb{Z}, n \neq 0 $ and $P(\xi = \frac{1}{n}) = \frac{p(1-p)^{n-1}}{2} , P(\xi = -\frac{1}{n}) = \frac{p(1-p)^{n-1}}{2}$, could be the solution of this task. But I don't know how to prove that it's returnable.