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- Imagine theres a hotel with infinite rooms. Theyre numbered one, two, three, four, and so on forever. This is the Hilbert Hotel and you are the manager. Now it might seem like you could accommodate anyone who ever shows up, but there is a limit, a way to exceed even the infinity of rooms at the Hilbert Hotel. To start lets say only one person is allowed in each room and all the rooms are full. There are an infinite number of people, in an infinite number of rooms. Then someone new shows up and they want a room, but all the rooms are occupied. So what should you do? Well, a lesser manager might turn them away, but you know about infinity. So you get on the PA and you tell all the guests to move down a room. So the person in room one moves to room two. The one in room two moves to room three, and so on down the line. And now you can put the new guest in room one. If a bus shows up with a hundred people, you know exactly what to do just move everyone down a hundred rooms and put the
