Show that a finite group of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.

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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