Some questions about proof of Theorem 2.43 in Baby Rudin

Question

I will include the proof here and highlight the parts that are giving me trouble.

Theorem $\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable.

Proof $\hspace{5 pt}$ Since $P$ has limit points, $P$ must be infinite. Suppose $P$ is countable, and denote the points of $P$ by $\mathbf{x_1}, \mathbf{x_2}, \mathbf{x_3}, \ldots$. We shall construct a sequence $\{V_n\}$ of neighborhoods as follows.

Let $V_1$ be any neighborhood of $x_1$.

1) ^ Are we using the Axiom of Choice here? How can we have an arbitrary set in a construction?

If $V_1$ consists of all $\mathbf{y} \in \mathbb{R}^k$ such that $|\mathbf{y} - \mathbf{x_1}| < r$, the closure $\overline{V_1}$ of $V_1$ is the set of all $\mathbf{y} \in \mathbb{R}^k$ such that $|\mathbf{y} - \mathbf{x_1}| \leq r$.

2) ^ It makes sense intuitively, but how do we prove this last statement?

Suppose $V_n$ has been constructed, so that $V_n \cap P$ is not empty. Since every point of $P$ is a limit point of $P$, there is a neighborhood $V_{n+1}$ such that (i) $\overline{V_{n+1}} \subset V_n$, (ii) $x_n \notin \overline{V_{n+1}}$, (iii) $V_{n+1} \cap P$ is not empty. By (iii), $V_{n+1}$ satisfies our induction hypothesis, and the construction can proceed.

3) ^ I really don't get this whole paragraph much at all. Could someone explain it in a more step-by-step way?

Put $K_n = \overline{V_n} \cap P$. Since $\overline{V_n}$ is closed and bounded, $\overline{V_n}$ is compact.

4) ^ "closed" comes from it being a closure and "bounded" comes from the definition of neighborhood, correct?

Since $x_n \notin K_{n+1}$, no point of $P$ lies in $\bigcap_1^\infty K_n$. Since $K_n \subset P$, this implies that $\bigcap_1^\infty K_n$ is empty. But each $K_n$ is nonempty, by (iii), and $K_n \supset K_{n+1}$, by (i); this contradicts the Corollary to Theorem 2.36.

Answer 1

$1):$ I don't think AC is used here. We just look at all possible neighborhoods of $x_1$ and pick one. After that, the construction can proceed.

$2):$ If $(y_n)$ is a sequence in $V_1$ and $y_n\rightarrow y$, then for all $\epsilon>0$ there is an $y_n$ such that: $|x_1-y|\le |x_1-y_n|+|y_n-y|\le r+\epsilon$. Then we must also have $|x_1-y|\le r$.

$3):$ Since $x_n$ is a limit point of $P$ and $V_n$ a neighborhood of $x_n$, there is a $p\in V_n\cap P$ such that $p\neq x_n$. Since $V_n$ is open, there is an $\epsilon$ such that $B_{\epsilon}(p)\subset V_n$. Put $\delta:= \min \left\{\frac{\epsilon}{2},\frac{|p-x_n|}{2}\right\}$. Then $x_n\not\in \overline{B_{\delta}(p)}$ and $\overline{B_{\delta}(p)} \subset V_n$. So we define $V_{n+1}:=B_{\delta}(p)$. Since $p\in V_{n+1}$, $p$ is a limit point of $P$ and $V_{n+1}$ is open, we have $V_{n+1}\cap P\neq \emptyset$.

$4):$ Yes, the sets are closed and bounded since they are closed balls with radius $r$.

Answer 2

My proof here includes a step-by-step construction of $V_{n+1}$ for step 3:

Proof of Baby Rudin Theorem 2.43

You might find it helpful. Step 3 is what most people (myself included) found puzzling about 2.43 when they first encountered it.