Example 1: Hilbert's Hotel is described in Wikipedia as following: Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room.
Example 2: Transforming it to a sequential moving Hotel: New guest arrives.
Step 1: guest 2 moves out room 2.
Step 2: guest 1 moves in room 2.
Step 3: guest 3 moves out room 3.
Step 4: guest 2 moves in room 3.
...
In the thought experiment example 2, you could describe it like this: guest 1 moves in 60 seconds, guest 2 in 30 seconds, guest 3 in 15 seconds and so on. After 2 minutes every guest moved from his current room to room n+1.
In example 1 an infinite amount of guests moved simultaneously. In example 2, all guest moved after 2 minutes.
What are the reason, why you think example 1 / example 2 is more faithful or both same faithful?
Let’s say every guest moves simultaneously his current room to room n+1. First I order all guests to stay out their rooms. Guest 1 stays before room 1, guest 2 before room 2 and so on. Then I will imagine what could happen. Guest 1 will move to room 2 and force guest 2 to move out his room. Guest 1 will move into room 2. Guest 2 has no room. Guest 2 will move to room 3 and force guest 3 to move out his room. Guest 3 hast no room and so on. The speed of imagination has no limit. For the first step my brain needed 1 minute, the second step could be done in 30 seconds. The third step in 15 and fourth in only 7.5. After two minutes, I caught every single step in my imagination. Now I let all infinite guests move simultaneously. For me my imagination should go the same way as moving all guests simultaneously, at least I did not find a reason why not. Case closed, question can be deleted if wanted. I am also fine with justifying me wrong. Have a nice day.