5. (Hilbert's Grand Hotel) The point of this exercise is to describe situations that can be resolved for infinite sets but would be impossible to address with finite sets. (Suggested reading: Chapter 2.5.2, example 2 is particularly useful.) The Grand Hotel has countable many rooms, numbered 1,2, 3, ..., and it is fully occupied. 1 (a) (6 points) Suppose that k guests arrive at the hotel. Show how you can accommodate all new guests without removing any of the current guests. (b) (6 points) Suppose Hilbert (the owner of the Grand Hotel) expands the property to a second building, also with infinite rooms numbered 1, 2, 3, .... Show how you can spread the current guests to fill every room in both buildings. (c) (8 points) (*) Suppose that a countable infinite number of guests arrive at the hotel. Show how you can accommodate all new guests without removing any of the current guests. 5. (Hilbert's Grand Hotel) The point of this exercise is to describe situations that can be resolved for infinite sets but would be impossible to address with finite sets. (Suggested reading: Chapter 2.5.2, example 2 is particularly useful.) The Grand Hotel has countable many rooms, numbered 1,2, 3, ..., and it is fully occupied. 1 (a) (6 points) Suppose that k guests arrive at the hotel. Show how you can accommodate all new guests without removing any of the current guests. (b) (6 points) Suppose Hilbert (the owner of the Grand Hotel) expands the property to a second building, also with infinite rooms numbered 1, 2, 3, .... Show how you can spread the current guests to fill every room in both buildings. (c) (8 points) (*) Suppose that a countable infinite number of guests arrive at the hotel. Show how you can accommodate all new guests without removing any of the current guests