When is full not full?

Hilbert's Grand Hotel has an infinite number of rooms.  Every room is occupied. Along comes someone who would like to stay there.  He can't be accommodated, yes? The hotel is full.  Not so, says Hilbert (the owner).  All I have to do is move the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, the guest in Room 3 to Room 4 and so on.  Because the hotel has an infinite number of rooms, there will always be a "next room" for each guest to move into, and Room 1 is now available for the newcomer.  I have to say I have problems with this, but what do you think? More later.