Hilbert Hotel

Here at the Hotel de Hilbert, there are always vacancies!

In 1925 Hilbert presented an infinity paradox that became better known as the Hilbert Hotel.

There are infinite guests in this hotel and it is packed, but whenever a customer arrives the manager asks guests to jump from room to room next door. Like this:

The guest of the room 1 jump to the room 2
The guest of the room 2 jump to the room 3

The guest of the room no jump to the room n + 1

So the paradox is that although it is always crowded, there are always vacancies at the Hotel de Hilbert.