I am a layman in this field so my understanding of the problem of "Hilbert's Hotel" is limited to the popular version presented to the public.
We know that Hilbert's Hotel can accommodate any finite number of guests; if one more guest arrives, we simply move the guest in room $1$ to room $2$, the guest in room $2$ to room $3$, and so on. If an infinite number of guests come, we move the guest in room $1$ to room $2$, the guest in room $2$ to room $4$, the guest in room $3$ to room $6$, and so on.
However, what happens if we have an infinite number of people leaving the hotel? Is every room still occupied? One way I thought about it is to work backwards from what we do if we add an infinite number of guests to the hotel. If the guest in room $3$ leaves, we just move the guest from room $6$ to room $3$, but.....oooops! The guest in room $6$ is already gone!