Situation: There's a hotel owner David Hilbert who owns a hotel with countably many (infinity that can be mapped by natural number surjectively) rooms, and there are countable guests who lived inside starting from room NO:1 and so on.
Now if a guest arrived…I guess we all know what to do–ask everyone to move over to the next door and spare Room NO:1 for the guest.
Now if a bus of countably new guests arrived…we can ask the residents to move to rooms with room number double of their current room number so as to save all the odd-numbered room for the new guests.
Question: But now what if countably many buses each with countably many guests arrived? What can we do to find new rooms for the new guests?
I know that the union of countably many countable sets is countable, and so far I am thinking about something to do with prime number factorization raise to the power of the number of their buses…but then how do I ask the occupants to move…?
Any thoughts or better room-assigning scheme?
Best Answer
Since academic salaries are not generous, Hilbert likes his hotel to have full occupancy.
Assume the hotel rooms are numbered $1$, $2$, $3$, and so on.
Move the guest currently occupying room $k$ to room $2k-1$.
The $k$-th person in bus $1$ goes to room $2^1(2k-1)$.
The $k$-th person in bus $2$ goes to room $2^2(2k-1)$.
In general, the $k$-th person in bus $j$ goes to room $2^j(2k-1)$.
We have full occupancy again.