Give an example of a sequence of real numbers with subsequences converging to every real number

Im unsure of an example Give an example of a sequence of real numbers with subsequences converging to every real number

Solution 1:

HINT: The rationals are a countable set, so they can be enumerated, and every real is the limit of a sequence of rationals.

Solution 2:

It's a bit of work, but you may show that taking the sequence $$a_n = n \bmod{2\pi}$$ has a limit point at every point in the interval $[-\pi, \pi]$, where the endpoints points are never actually attained attained (otherwise $\pi$ would be rational). So you may continuously map the interval $(-\pi,\pi)$ into $\mathbb{R}$ in you favorite way. Take, for instance, the map $f: (-\pi,\pi) \to \mathbb{R}$ sending $x \mapsto \tan(x)$. So now you have a sequence $a_n$ converging to every point $a \in (-\pi,\pi)$, then since tangent is continuous, $\tan(a_n) \to \tan(a)$, and $\tan$ is surjective onto $\mathbb{R}$, so you are done.

I realize that the first part, taking integers modulo $2\pi$ is not entirely necessary for the proof, since $\tan(x) = \tan(x + 2k\pi)$, but I think it's easier to break it into smaller problems.