Hotel infinity
You are the manager of the Hotel Infinity, a hotel that has infinitely many rooms. No matter how many people does the hotel occupy, you can always add new guests to it. However, when you are just going off-duty, a group of people arrive for a convention. There are an infinite amount of new guests in the convention. If you already have an infinite number of people in the hotel before the convention, it is possible to accommodate the new guests?
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Number the rooms from 1 to infinity. The person in room x moves to room 2x, and so there are infinity empty odd-numbered rooms for the infinity guests.