Suppose that Hilberts Grand Hotel is fully occupied but the

Suppose that Hilberts Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel. Show that a countably infinite number of guests arriving at Hilberts fully occupied Grand Hotel can be given rooms without evicting any current guest.

Solution

1) Now consider below to solve the problem

After closing all even numbered room we are left with {1,3,5...} which if we generalise 2k+1, now available rooms are 2k+1 number of rooms. Now let see further arrangement, it is possible to shift all the guest in odd numbers to odd number rooms where k is an even number.and having this arrangement all the odd room numbers with k as odd numbers are free.Guests in even numbered rooms n move to 2 (n 1) + 1. Guests in odd numbered rooms n = 2k + 1 move to 2*2k + 1