In the Understanding **Hilbert’s Grand Hotel**, we have discussed the brilliant schemes of a hotel manager in accommodating finite and infinite number of guests in a hotel with infinite number of rooms, where each room was occupied by one guest. In other words, the hotel was fully occupied. In this post, I will explain the mathematics behind these schemes. To be able to understand the explanation, it his highly recommended that you read first the post in the link above.

#### Finite Number Of Guests

In the Grand Hotel problem, during the first night, a guest arrived. The hotel was full, so there was no room available. However, to accommodate the new guest, the manager requested the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, the guest in Room 3 to move to room 4 and so on. This means that each guest had to move to the room whose number is 1 higher than the the current room number. This leaves the Room 1 vacant.

Now, how is this possible? » Read more