Hilbert's Hotel

Hilbert’s Hotel is a (hypothetical) hotel with an infinite number of rooms, each one of which is occupied. The hotel gives rise to a paradox: the hotel is full, and yet it has vacancies.

Suppose that a new visitor arrives; can he be accommodated? At first it seems that he cannot, but then the hotel clerk has an idea: He moves the guest in Room 1 to Room 2, and the guest in Room 2 to Room 3, and so on. Every guest is moved to the next room along.

For every guest, in every room, there is another room into which they can be moved. This leaves Room 1 vacant for the new visitor. Although the hotel is full, then, the new guest can be accommodated in Room 1.

By moving every guest to the room the number of which is double the number of their current room, all of the odd numbered rooms can be vacated for new guests. There are, of course, an infinite number of odd numbered rooms, and so an infinite number of new guests can be accommodated.

Quote: I've always thought a hotel ought to offer optional small animals. I mean a cat to sleep on your bed at night, or a dog of some kind to act pleased when you come in. You ever notice how a hotel room feels so lifeless?” - Anne Tyler