Hilbert's "Infinity hotel"

I am loving this: Amanda Boyle's short film Hotel Infinity concerns a hotel with an infinite number of rooms. Its slogan is "We're always full, but we always have room for you."

Guffaw guffaw!

Now let's get onto the serious stuff:
1) In order to check a new guest into the infinity hotel, you need to get the guest in room 1 to move into room 2 and room 2 into room 3 and so on. Then you will free up one room in front for the new guest. You cannot check in the new guest into the last room because there is no last room and every room is already occupied.

Infinity + 1 = infinity

2) In order to check an infinite amount of new guests into the hotels, you need to move all existing guests to the room which is double the room number they have already (ie the person in room 4 moves to room 8; the person in room n moves to room 2n). The first person in the newly-arrived group gets room 1; the nth person gets room 2n-1. In this way, an infinite number of odd-numbered rooms get freed up.

Infinity + Infinity = Infinity

3) If an infinite amount of guests leave the next morning, how many guests would be left? The answer is interesting. It could be that any real numbers of guests stayed behind while the infinite rest checked out or it could be that all the guests checked out.

Infinity - Infinity = Any real number (1,2,3 etc) OR 0