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 ?
 A: 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.)
A: -- For $\,n=1,3,5,..., 2n-1,...\,$ , define $\,a_n=1\,$
-- For $\,n=2,6,10,...,2+4(n-1),...\,$ , define $\,a_n=2\,$
-- For $\,n=4,12,20,..,4+8(n-1),...\,$ , define $\,a_n=3\,$
...............................
-- For $\,n=2^k,2^k+2^{k+1},...,2^k+2^{k+1}(n-1),...\,$ , define $\,a_n=k+1$
...............
Now just take the sequence $\,\left\{a_n\right\}_{n=1}^\infty\,$
