"Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied." (*)
To say that the hotel is fully occupied is to say that, for all n in N, room n is occupied. Therefore, room 1 is occupied and, for all n in N, if room n is occupied, room n+1 is also occupied. Hence, for all n in N, it is not the case that a guest can move from room n to room n+1. Namely, no more guests can be accommodated. QED.