 Hilbert’s Hotel has infinitely many rooms, labelled 1, 2, 3, …, each of which is currently occupied by a guest. Despite the fact that the hotel is already full, a new guest who turns up at reception is readily accommodated: for each n, the guest in room \(n\) is moved to room \(n+1\), and the new guest is installed in room 1. Indeed, despite the fact that the hotel is already full, it can accommodate infinitely many new guests: for each \(n\), the guest in room \(n\) is moved to room \(2n\), and the new guests are installed in the odd-numbered rooms. Of course, if the infinitely many people in the odd-numbered rooms check out, there are infinitely many people left in Hilbert’s Hotel; but if infinitely many people check out from all but the first three rooms, only three people remain.