Let $C\subset \mathbb{R}$, $C$ infinite. Suppose that there exist a family of compact sets $\{Q_k\}_{k\in \mathbb{N}}$ such that:
$Q_{k+1}\subseteq Q_k$
$Q_k\cap C$ is infinite $\forall k.$
By the nested segments intervals theorem, I know that $Q:=\bigcap_{k\in \mathbb{N}}Q_k\neq \emptyset$. I want to know if also is true that $Q\subseteq C$ or at least $Q\cap C\neq \emptyset$. Thanks by your help.
Edit: Assume that $C$ is a perfect set.