Hotel Infinity

I heard about this from Wen Cin a few weeks ago. After reading more about it, I have decided to write a short post about it in my blog. However, I am not good enough in maths to understand this thoroughly so please forgive me for any inaccuracy in my post.

Lets assume there is a hotel of infinitely many rooms and every room is occupied.
Case 1: A new guest comes and requests for accommodation.
Solution: Guest in room 1 shifts to room 2, guest in room 2 shifts to room 3 and so on ( meaning guest staying in room n shifts to room n+1 ). Then the new guest will stay in room 1

Case 2: A bus carrying a countable infinite number of guests arrive at the hotel and all of them request for accommodation.
Solution: Guest in room 1 shifts to room 2, guest in room 2 shifts to room 4 and so on (meaning guest staying in room n shifts to room 2n, leaving all the odd number room empty). Then all the new guests will accommodate the odd number room.

Case 3: A countable infinite number of buses, each carrying a countable infinite number of guests arrive at the hotel and all of them request for accommodation.
Solution: Guest in room 1 shifts to room 2, guest in room 2 shifts to room 4 and so on (meaning guest staying in room n shifts to room 2n, leaving all the odd number room empty). All the passengers from bus 1 will accommodate rooms 3ⁿ for n = 1,2,3 ... The passengers from bus 2 will accommodate rooms 5ⁿ , and so on ( meaning passengers from bus i will accommodate rooms pⁿ where p is the (i+1)-th prime number).

Source: Wikipedia