Paradox of Hilbert's Great Hotel

In 1925 Hilbert presented an infinity paradox that became better known as the Hilbert Hotel. In this hotel there are infinite rooms and it is always packed with one guest in each room. But whenever a customer arrives, the manager asks guests to jump from room to room next door. Like this:

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

The guest of bedroom n jump to room n + 1

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

There is, however, only one problem with this paradox. To transfer the guest from room 1 to room 2, room 2 must be free, but to be free room 3 must also be free to transfer the guest from room 2 to 3, and so on. Therefore, the waiting time to release room 1 would be infinite, since for this to happen, the nth room must be free to release room n-1.