Construct a sequence that satisfies the following property.

Question

I'm studying for a test tomorrow and I'm stuck on this question currently. Any advice would be much appreciated.

Construct a sequence {$x_k$} $\subseteq \mathbb{R}^2$ with the property that for any x $\in{\mathbb{R}^2}$, there exists a subsequence {$x_{k_n}$} which converges to x

Answer 1

Make your $\{x_{k}\}$ an enumeration of any countable dense subset of ${\bf R}^2$. For example, form $\{x_{k}\}$ by enumerating all the points in ${\bf R}^2$ with rational coordinates.