Cantor's intersection theorem for nested open sets.

Cantor's intersection theorem for nested open sets.

Let $\{I_k\}$ be a set of nested closed bounded sets such that $\forall
k\in\Bbb{N}, I_k\supset I_{k+1}$, and
$\lim_{n\to\infty}\operatorname{diam}(I_n)=0$. Let the metric space
$(X,d)$ be complete. This implies that $\bigcap_{k=1}^\infty
I_k\neq\emptyset$- it contains precisely one point.
Let $J_k=\operatorname{Int}(I_k)$, or the interior of $I_k$. What can we
say about $\bigcap_{i=1}^\infty J_k$?
I will list down my assertions one by one:
The infinite intersection of closed sets is a closed set. This is because
it is the complement of the infinite union of open sets, which is an open
set, and the complement of an open set is a closed set.
So $S=\bigcap_{k=1}^\infty I_k$ is a closed set containing one point. Let
that one point be $x$. $x$ can't be the limit point of
$\operatorname{Int}(S)$, as $S\setminus \{x\}$ is empty. Hence,
$\bigcap_{k=1}^\infty J_k=\bigcap_{k=1}^\infty I_k=\{x\}$, which is a
closed point, but not an open point.
Are the arguments sound?
Thanks in advance!