TITLE: Accumulation points: an example of a sequence that has exactly two accumulation points
QUESTION [5 upvotes]: The definition in my text defines A.P. as: A point p is an A.P. of a set S if it is the limit of a sequence of points of S-{p}.
I am confused by my classnotes and information about A.P. in my textbook is not enough for me to understand the topic.

REPLY [4 votes]: The sequence $(-1)^n\left(1+\frac{1}{n}\right)$ has exactly two A.Ps $1$ and $-1$.
More generally if $(a_n)_{n\in \mathbb{N}}$ is a sequence s.t. $a_n \to a \neq0 $ and the set $\{a_n:n \in \mathbb{N}\}$ is infinite then the sequence $((-1)^na_n)_{n\in \mathbb{N}}$ has exactly two accumulation points $a$ and $-a$.