I have a question. I thought I knew the answer, but for some reasons I can't see how to prove it anymore. Suppose you have a countable dense subset D of $\mathbb R^2$. I want to construct another dense subset of $\mathbb R^2$ with certain properties. From what I remember, if I make sure to choose $b_n\in\mathbb R^2$ such that $d(a_n, b_n) < 1/n$, then the set $\{b_1, b_2, \ldots,\}$ will be dense in $\mathbb R^2$. But I do not remember how to prove this anymore.
Thank you for your help.