Moving Guests to occupied rooms??? 
 -'Because the hotel has infinite rooms, we can move any guest occupying any room n to room n+1, then fit the newcomer into room 1'. 
 No, we can't. Room n+1 is already occupied, as are all rooms by definition. Any room in the hotel has 'occupied' as descriptor, as per condition of the problem (' a hypothetical hotel with a countably infinite number of rooms, all of which are occupied '). So, whatever number we call any particular room (be it n+1, n+2, n+3 or any other), it does not change the room's 'occupied' status. Therefore, we can't move a guest from an occupied room to an occupied room to make room for more guests )