Is it possible to find a sequence $\{a_n\} \in \mathbb{R}^n$ with the property that for any $x \in \mathbb{R}^n$, $\{a_n\}$ has a subsequence that converges to $x$?
I tried approaching it by first finding a sequence in $\mathbb{R}$ with the above property. Consider a sequence $\{b_n\} \in \mathbb{Q}$ that contains every rational number an infinite number of times. It should be possible to construct such a sequence because the rationals are countable. We also know by the construction of $\mathbb{R}$ that every real number $x$ has some sequence $\{x_i\} \in \mathbb{Q}$ that converges to it. And as each rational number will appear within $\{b_n\}$ an infinite times, we can certainly find $\{x_i\}$ as a subsequence of $\{b_n\}$.
I'm stuck when it comes to generalizing this process to $\mathbb{R}^n$. Perhaps we could consider all possible n-tuples of rational numbers and put these in a sequence so that each shows up an infinite number of times? Is that even countable? I'm unsure as to how I can state this rigorously. I'm only looking for a little insight, as I'd like to see the solution for myself.