Question

Suppose that a countably infinite number of buses, each containing a countably infinite number of guests, arrive at Hilbert’s fully occupied Grand Hotel. Show that all the arriving guests can be accommodated without evicting any current guest.

Short answer

Move the guest currently Room i to Room for 2i + 1 for i =1, 2, 3, ... Put the $j^{th}$guest from the $k^{th}$ bus into Room $2^k(2j+1)$

Step-by-step solution

Step: 1

Move the guest currently Room i to Room 2i + 1 for i = 1, 2, 3, ... Put the guest from the $k^{th}$ bus into Room$2^k(2j+1)$