Exercise: Let $S_n$ for $n \geq 1$ be a finite union of disjoint closed balls in $R^k$ of radius at most $2^{-n}$ such that $S_{n+1} \subset S_n$ and $S_{n+1}$ has at least 2 balls inside each ball of $S_n$. Prove that C = $\cap_{n \geq 1} S_n$ is a perfect, nowhere dense compact subset of $R^k$.
I've proved compactness and set being nowhere dense, but I fail to see how it is perfect. The hint in the exercise is to use Cantor set construction for aid.
Let $x\in C$. Then $x\in S_n$ for each $n\in\mathbb{N}$. For each $n$, let $B_n$ be the closed ball of $S_n$ that contains $x$.
Claim. Each $B_n$ contains a point of $C$ other than $x$.
Proof. For fixed $n$, let $B$ be a closed ball of $S_{n+1}$ contained in $B_n$, that does not contain $x$(possible since there are at least two balls inside $B_n$). Then we show that $B$ contains a point of $C$. Inside $B$, we can choose nested closed balls $B'_m$ of $S_m$, for $m\ge n+1$, and there intersection $\cap_{m=n+1}^{\infty}B'_m\subset\cap_{m=n+1}^{\infty}S_m$ is nonempty, and the point is a point of $C$. This is not $x$ since $x\notin B$. $\blacksquare$
Then let $x_n\in B_n\cap C\setminus\{x\}$. Since $x, x_n\in B_n$, $|x_n-x|\le 2\cdot 2^{-n}\to 0$, so $x_n\to x$.