In this case, we first must revisit the paradox of Hilbert’s Hotel.
“HILBERT’S GRAND HOTEL: We now describe a paradox that shows that something impossible with finite sets may be possible with infinite sets. The famous mathematician David Hilbert invented the notion of the Grand Hotel, which has a countably infinite number of rooms, each occupied by a guest. When a new guest arrives at a hotel with a finite number of rooms, and all rooms are occupied, this guest cannot be accommodated without evicting a current guest. However, we can always accommodate a new guest at the Grand Hotel, even when all rooms are already occupied.” Discrete Mathematics and its Applications by Rosen.
Following the description above, let’s solve the exercise.
Let’s n be the number of new guests that arrive at the hotel.
If we move the guest in room 1 to room n+1, the guest in room 2 to room n+2, the guest in room 3 to room n+3, and so on until we move all guests in rooms 1 to n.
Then, we can assign the first n rooms to each of the new guests without evicting any current guests.
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