%
\label{prop:cylinders-are-balls}
For every $x \in \tau^3$ and every stage $k \geq 0$,
the stage-$k$ cylinder $C_k(x)$
(Definition~\ref{def:stage-k-cylinder})
is simultaneously a closed ball and an open ball
in the ultrametric~$d$:
\[
    \boxed{C_k(x)
    \;=\;
    \overline{B}(x, 2^{-k})
    \;=\;
    B(x, 2^{-(k-1)})}
\]
where $\overline{B}(x, r) = \{y : d(x,y) \leq r\}$
is the closed ball
and $B(x, r) = \{y : d(x,y) < r\}$
is the open ball.

By Definition~\ref{def:stage-k-cylinder},
$y \in C_k(x)$ if and only if
$\pi_k(y) = \pi_k(x)$,
which holds if and only if
$\delta(x,y) \geq k$.

\emph{Closed ball.}
$\delta(x,y) \geq k$ is equivalent to
$d(x,y) = 2^{-\delta(x,y)} \leq 2^{-k}$,
so $C_k(x) = \overline{B}(x, 2^{-k})$.

\emph{Open ball.}
Since $d$ takes values in $\{0\} \cup \{2^{-n} : n \geq 0\}$
and $2^{-(k-1)}$ is the next value above $2^{-k}$
in this discrete set,
the condition $d(x,y) \leq 2^{-k}$
is equivalent to $d(x,y) < 2^{-(k-1)}$.
Hence $C_k(x) = B(x, 2^{-(k-1)})$.