Why does a finitely oscillating sequence always has at least $2$ limit points? **Why** is it not possible to find a finite oscillating sequence which has only $1$ limit point or no limit point at all? Why must it always have at least $2$? I'm just not able to picture it. Thanks **EDIT** : I think I should've started out by saying how do you even picture a finite oscillating sequence? What is an intuitive way to imagine a finite oscillating sequence?

We can always say that there always exist at least one limit point of a bounded sequence (why??)[ If It's infinite then we can apply Bolzano weierstrass theorem, otherwise it must be finite and must have a limit point].

Let $(a_n)$ be a finite oscillating sequence. Suppose there exists a limit point say $\alpha$ and since $(a_n)$ is not convergent implies there exists a subsequence which doesn't converge to $\alpha$ and the subsequence thus obtained is also a bounded subsequence, again by Bolzano Weierstrass theorem we get a new limit point.

Hence $(a_n)$ has at least 2 limit points