Bounded sets of isolated points in compact metric spaces

Bounded sets of isolated points in compact metric spaces

Context and definitions:
Say $(X,d)$ is a compact metric space, with $f: X \rightarrow X$
continuous. For each $n \in \mathbb{N}$, the metric
$d_{n}(x,y) = \max_{0 \leq k \leq n-1}d(f^{k}(x),f^{k}(y))$
measures the maximum distance between the first $n$ iterates of $x$ and
$y$. Let $\epsilon > 0$. A subset $A \subseteq X$ is
$(n,\epsilon)$-$separated$ if any two distinct points in $A$ are at least
$\epsilon$ apart in the $d_{n}$ metric.
My questions is about the statement: "Any $(n,\epsilon)$-separated set is
finite."
My work so far:
Suppose $A \subseteq X$ is $(n,\epsilon)$-separated. Each point of $A$ is
isolated in $A$, thus $A$ is closed. Since we are in a compact space, $A$
is compact. Since the space is a metric space, $A$ is bounded. My
intuition is that this, along with $A$ being $(n,\epsilon)$-separated,
means that $A$ is finite. This looks to be true in some simple
spaces($\mathbb{R}^{2}$, for example), but I'm afraid that for a space
that has too many "degrees of freedom," this might not be true.. Thanks
for any advice.