Puzzles/Set theory puzzles/Infinite Hotel

Puzzles | Set theory Puzzles | The infinite hotel

You are the manager of a hotel with an infinite, countable number of rooms. All rooms are currently being occupied. One day a new guest arrives and asks for a room.
 * How do you make room to accommodate this new guest?

Later on that day, the infinite hotel next door has a flood and all of those guest come over to your hotel and ask for a room.
 * A countable, infinite number of guests arrive. Can you make room for them all?

Solution