Suppose that Hilbert’s 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.

Step 1 :

In this problem in the Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance then we have to prove that all guests can remain in the hotel.

Given in the problem by closing even numbered rooms,

Then we have the rooms number {1,3,5……}

All available room numbers are in the form of ,

Step 2 :

It is possible to move all the guest in even and odd room numbers to odd room numbers where k is an odd numbers are free.

.’. Guests in even numbered rooms move to .

Guests in odd numbered rooms move to