Example of a sequence with at least 3 limit points

Question

What is an example of a sequence that has at least 3 limit points?

Answer 1

$x_n=\sin \frac{n\pi}2$ has limit points $0,1,-1$.

For any $m\in\Bbb N$, $x_n = n\bmod m$ has $m$ limit points $0,1,\ldots,m-1$.

$x_n=n \bmod {\lfloor \sqrt n\rfloor^2}$ has countably many limit points (namely $\Bbb N_0$).

$x_n=\sin n$ has uncountably many limit points, namely $[-1,1]$

We know that $\Bbb Q$ is countable. If $x_n$ is an enumaration of $\Bbb Q$, any element of $\Bbb R$ is a limit point.