Suppose you have a hotel which has one floor with infinite number of rooms in a row and all of them are occupied. Hint
1) A new customer wants to check in, how will you accommodate her?
2) What if infinite number of people want to check in, how will you accommodate them?
3) Suppose infinite number of buses arrive at the hotel, each having infinite number of people, how will you accommodate them?
1) Since there are infinite number of rooms and infinite+1= infinite
Just ask person in room k to move to k+1, thus making the first room vacant. :)
2) In the other case, since infinite+infinite = infinite
asking person in room k to move to 2k solves the problem.
3) Since NxN is countable set. We can get a 1-1 mapping from N to NxN
Hence, we can accommodate (infinite people X infinite buses) in the hotel.