Constructing sequence whose subsequence converges to given point always exists

Question

How to construct a sequence in R^2 such that for any x in R^2, there exist a subsequence converging to x ?

Answer 1

Note that $\mathbb Q^2$ is dense in $\newcommand{\epi}{\twoheadrightarrow}\mathbb R^2$, so all we need is a surjection $\mathbb N\epi \mathbb Q^2.$ It might be tricky to write it directly, so I'll just break it into simpler problems:

$$\mathbb N\to\mathbb N^2\to \mathbb N^4 \to \mathbb Z\times\mathbb N\times \mathbb Z\times\mathbb N \to \mathbb Q^2.$$

I leave it to you as an exercise to fill the arrows. One more hint: you need bijections $\mathbb N\to\mathbb N^2$ and $\mathbb N\to\mathbb Z$.