Hilbert's Grand Hotel

We talked over dinner about Hilbert's Grand Hotel (follow the link for a good description). The hotel has an infinite number of rooms, all full, and a new guest arrives. To make room for a new guest, for all i the guest in room i is moved to room i+1.

My 12-year-old daughter argued that the hotel is too expensive to build. I postulated that room i+1 costs only half as much as room i, and that the guests shrink as they are moved to higher numbered rooms. She conceded that such a hotel could exist.

My 15-year-old daughter argued that the moving of guests fails. For example, what happens to the guest in room number infinity? I said "there is no room numbered infinity". She responded, "then what number is it?", and the whole family felt that she won.