"Since $\mathbf{x}_n \notin K_{n+1}$, no point of $P$ lies in $\bigcap_{1}^\infty K_n$".
As far as I understand, $K_n$ is a compact subset of a perfect set $P$ which is assumed as a countable set, $P = \{\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3,\dots\}$, and $K_{n+1}$ is constructed such that $\mathbf{x}_n \not\in K_{n+1}$ and $K_{n+1} \subset K_n$. The proof ends by showing the contradiction that $\bigcap_1^\infty K_n$ is both empty and nonempty.
I have a problem in understanding the sentence in the quotation marks. If we assume $P = \{1,1/2,1/3,\dots\}\cup\{0\}$ and define $K_n = [1/n,0]$, then clearly $\bigcap_{1}^\infty K_n = \{0\}$. What am I missing here?