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A comment on Does infinity exist?. Parent comment
Because every natural number n has a successor n+1, you can do it for all of them.
Each n except 1 has a predecessor n-1 from which guests where moved to n. Nobody moved into 1, therefore you can put the new arrivals there.
After that, there are still (countably) infinite many guests. They are just matched up differently with the rooms/numbers.
Because you have infinitely many rooms, you were able to accomodate one more by matching them up differently. That's sort of the point :-)
Because every natural number n has a successor n+1, you can do it for all of them.
Each n except 1 has a predecessor n-1 from which guests where moved to n. Nobody moved into 1, therefore you can put the new arrivals there.
After that, there are still (countably) infinite many guests. They are just matched up differently with the rooms/numbers.
Because you have infinitely many rooms, you were able to accomodate one more by matching them up differently. That's sort of the point :-)