Problems in Theorem 2.43 of baby Rudin

For the 3rd paragraph:

$V_n = N_{\epsilon_n}(p_n)$ some $p_n \in P, \epsilon_n > 0$. $\exists$ infinitely many elements of $P$ in $V_n$, so pick one $p_{n+1}$ that is not equal to $x_n$ or $p_n$. Let $\epsilon=d(p_n, p_{n+1})$, $\epsilon'=d(p_{n+1}, x_n)$, $\epsilon''=\epsilon_n-\epsilon>0$. Choose $\epsilon_{n+1} < \min\left\{\epsilon, \epsilon', \epsilon''\right\}$, then:

• $d(x_n, p_{n+1}) > \epsilon_{n+1}$ so certainly $x_n \not\in \overline{V}_{n+1}$
• if $d(e, p_{n+1}) \leq \epsilon_{n+1}$ then $$\begin{align}d(e,p_n) &\leq d(e,p_{n+1}) + d(p_n, p_{n+1}) \\ &\leq \epsilon_{n+1} + \epsilon \\ &< (\epsilon_n - \epsilon) + \epsilon = \epsilon_n\end{align}$$ so $\overline{V}_{n+1} \subset V_n$.
• $p_{n+1} \in V_{n+1} \cap P$ so intersection is non-empty.