Original: But the hotel of interest has infinitely many rooms, so if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1.
Simplified: First, 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, the guest in room 4 to room 8, and so on.