I'm having a great difficulty understanding a proof of a lemma from this paper:
http://www.math.hawaii.edu/~erik/papers/cat0-A.pdf
It's lemma 1.12. To make it shorter for anyone who'd like to take a look: we are considering $ X $ - a $ \text{CAT(0)} $ cube complex. The set $ \mathfrak{H}(x,y) $ is a set of hyperplanes which separace vertices $ x,y $.
The moment I'm lost is the sentence: ,,since $ \mathfrak{H}(x,y) $ is finite...''. I have no idea why it might / should be true.
I'd appreciate any form of hint / explanation