$p$ is a cluster point of $S\subset M$ if each neighborhood of $p$ contains infinitely many points. Here is my confusion, a cluster point is also a limit point of $S$, right?
If so, then how does the sequence $((-1)^n)$, ${n\in \mathbb N}$ has two cluster points namely $1, -1$ especially since the sequence does not have a limit as n approaches infinity.