Cluster point of the sequence $a_n = (-1)^n$

Question

I am trying to understand an example given of the cluster points of the sequence $a_n=(-1)^n$.It is given that -1 and 1 are the only cluster points of this sequence.But i am unable to understand mathematically why they(-1 &1) are the cluster point.The definition of the cluster point that i know of is as follows: A real no. $p$ is said to be a cluster point of a sequence <$a_n$> if every interval around $p$ contains infinitely many points of <$a_n$>

Answer 1

For example, for all $\epsilon>0,$ $(1-\epsilon,1+\epsilon)$ contains all $a_n$ for $n$ even, which is infinitely many terms. But take $p=1-\epsilon,$ then the interval $(p-\epsilon/3,p+\epsilon/3)$ contains no points of $\{a_n\}$ so is not a cluster point.