Example of a sequence with countable many cluster points Can someone give a concrete example of a sequence of reals that has countable infinite many cluster points ?

First fix some bijection $f : \mathbb{N} \to \mathbb{N} \times \mathbb{N}$.  For $n \in \mathbb{N}$ let $g(n)$ denote the first coordinate of $f(n)$ and let $h(n)$  denote the second corrdinate.
Then define a sequence $\{ x_n \}_{n=1}^\infty$ by $$x_n = g(n) + 2^{-h(n)}.$$
Then every natural number is a cluster point of this sequence.
For a more concrete example, consider the following sequence:
$$\begin{array}{c|c}
n & x_n \\
\hline
1 & 1 + 2^{-1} \\
2 & 1 + 2^{-2} \\
3 & 2 + 2^{-1} \\
4 & 1 + 2^{-3} \\
5 & 2 + 2^{-2} \\
6 & 3 + 2^{-1} \\
7 & 1 + 2^{-4} \\
8 & 2 + 2^{-3} \\
9 & 3 + 2^{-2} \\
10 & 4 + 2^{-1} \\
\vdots & \vdots
\end{array}
$$
(I'm too lazy to give an exact formula at the moment, but I think the idea is clear.)
