Guests leaving Hilbert's Hotel?

Question

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!

Answer 1

As mentioned in the comments, it depends on which guests leave. Depending on which guests leave, the number of guests remaining can be any finite number, or infinity. In other words, "$\infty - \infty$" isn't well-defined.

This is related to the following puzzle. Initially you have some dollars. The Devil comes to you and tells you that for every positive integer $n$, when the clock strikes $\frac{1}{2^n}$ minutes before midnight, he will give you two dollars and take away a dollar from you. How much money do you have at midnight?

The curious answer is that while the amount of money you have is increasing up until midnight, at midnight it can be infinite or any finite amount, including zero, depending on which dollars the Devil took from you. In particular it's possible for the Devil to take all of your money.