While demonstrating importance of infinity he gave an example of a hotel with infinite rooms. This Hotel is called Hilbert’s hotel. One night a guest, who is a mathematician, appears in the reception of the hotel and requests for a room. The receptionist tells him that all the rooms in the hotel are full. Generally hotel room numbers are natural numbers and start with 1, 2, 3 etc. The mathematician guest thinks for a while and tells the receptionist about possibility of giving him a room without asking any of the occupants to leave the hotel. Receptionist wondered and asked the mathematician to explain. The mathematician says that it is simple. You can ask occupants to shift to next room. For instance you can shift occupant of room no. 1 to room no. 2, room no. 2 to room no. 3 ….so on….and occupant of room no. (n) to room no. (n+1). Give me room no. 1. Since there are infinitely many rooms in Hilbert’s hotel this works.
Now, what is two guests appear in the Hilbert's hotel which is full and ask to get accommodated without asking previous occupants to move out?
What if a group of (n) guests appear in Hilbert's hotel which is full and ask to get accommodated without asking previous occupants to leave the hotel?
The occupant in the room no. 1 will be shifted to the room no. n+1, the occupant in the room no. n+1 will be shifted to the room no. n+n+1=2n+1 and so on. The occupant in the room no. 2 will be shifted to the room no. n+2. The occupant in room no. n+2 will be shifted to n+n+2=2n+2 and so on. The occupant in the room no. n will be shifted to the room no. n+n=2n. The occupant in room no. 2n will be shifted to room no. n+2n=3n and so on. Thus the first n rooms will be vacant and the n gusts could be accommodated.
I think there is a simpler, slightly different answer to your question: if there appear m guests (m being finite), all occupants are to move from their current room n to the room n+m, leaving the first m room available.
what if m is infinite? in that case, occupants are asked to move from room n to room 2n. After moving, all (infinitely many) odd rooms are available to the newcomers.
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