Imagine you are a manager of a very special hotel, a hotel with an infinite number of rooms, where each room has a natural number on the door \(1,2,3,4,...\). Only one guest can stay in each room and in most cases the hotel will be initially full with no vacant rooms left.
You will have to deal with unusual situations that may occur.
A new customer comes to the hotel and wants a room. It happened today that all the rooms are occupied. What should you do?
Now imagine you got \(10\) new guests arriving to the completely full hotel. What should you do now?
The next day you have even harder situation: to the hotel, where all the rooms are occupied arrives a bus with infinitely many new customers. In the bus all the seats have numbers \(1,2,3...\) corresponding to all natural numbers. How to deal with this one?
Imagine you have \(2\) new guests arriving to the full hotel. How do you accommodate them?
What would you do about \(10000\) new guests arriving to the full hotel?
Imagine you have now a general finite number of new guests arriving to the full hotel. What do you do?
Today you saw two infinitely long buses with seats numbered as \(1,2,3,...\) carrying infinitely many guests each arriving at the full hotel. How do you accommodate everyone?
Now there are finitely many infinitely long buses with seats numbered as \(1,2,3,...\) carrying infinitely many guests each arriving at the full hotel. Now what do you do?
How about infinitely many very long buses with seats numbered \(1,2,3...\), each carrying infinitely many guests, all arriving at the hotel. Assume for now that the hotel is empty. But that seems like a lot of guests to accommodate. What should you do?