Analysis WebNotes

Hilbert's Hotel

Imagine a hotel, which is much like any hotel you might have stayed at, with rooms, and numbers on the doors of each of the rooms. Unlike any hotel you have (most likely) stayed in, this hotel however has infinitely many rooms, and the rooms' numbers go through all of the natural numbers.

Now imagine that on a particularly busy night, the hotel is completely full. A bus-load of 50 tired, grouchy tourists turns up, and demand rooms. But there's been a double booking and their rooms are already taken! What's to be done?

Unperturbed the landlord presses a button on her desk once. An announcement goes out through the hotel asking every guest to move out of their room to a room fifty doors further down. Ten minutes later, when the commotion has settled down, there are fifty free rooms available, and everyone still has a room...

Question: Wouldn't the landlord have had a problem if the bus, too, had infinite capacity, and carried infinitely many tourists?