I'm looking at the association between reflection groups and Coxeter groups in Bourbaki (books 4-6), and I have a question about some implications of their conditions. Here, $\mathcal{H}$ represents a collection of hyperplanes in a finite dimensional Euclidean space, and $W$ is the group generated by the reflections in $\mathcal{H}$. We suppose that $W \mathcal{H} = W$. We say that $\mathcal{H}$ is locally finite if any compact set contains finitely many hyperplanes in $\mathcal{H}$. They state the following condition on $\mathcal{H}$:
$$ \forall K,L \text{ compact}, |\{ w \in W \mid wK \cap L \neq \emptyset \}| < \infty,$$
which they claim implies that $\mathcal{H}$ is locally finite (p. 77, Lemma 1). I am trying to prove the converse. That is, if $\mathcal{H}$ is locally finite, then the equation holds. I am proceeding by contraposition.
In the case that $A = \{w \in W \mid wK \cap L \neq \emptyset\}$ contains infinitely many reflections, it is clear to me that the closure of the convex hull is a closed and bounded set incident with every reflection in $A$. Therefore it is a compact set incident with infinitely many elements of $A$, and we have $\mathcal{H}$ not locally finite. It isn't so clear to me how to generalise this argument so that $A$ does not contain infinitely many reflections. In particular, the existence of compact sets with infinitely many connected components was annoying.