clusterpoints of periodic sequences

Question

is every element of a periodic sequences also a clusterpoint of that sequence? from the definition of clusterpoint, namely that a sequence has a subsequence that converges to this point couldn't one say that there are constant subsequences for all the points? take sine for example.

Answer 1

Let $(a_n)$ be a periodic sequence with period $p \ge 1$. Let $n_0 \in \mathbb N$ and define $(b_j)$ by

$b_j=a_{n_0+jp}$.

Then $(b_j)$ is a subsequence of $(a_n)$, which converges to $a_{n_0}$, since $b_j=a_{n_0}$ for all $j$.