Before you read this article further, I will tell you upfront that no, I am not going to add new mathematical insight. This is just a brief writing after a brief thought of infinite hotel.
For those of you who does not know “Infinite Hotel Paradox”, you should watch this video first
So, what do I mean by implication? First, as a physicist, we assume that information cannot travel faster than the speed of light. Thus, there is not way the receptionist can simultaneously make each occupant from room to the room. At best, the receptionist can nudge occupant in room and politely ask him/her to knock the door at room for them to vacate the room and move to room , and so on.
The receptionist cannot do all the information transmission by himself since it will take forever for him to do so (or else he cannot entertain the latest guest to check in). In the end, he can only rely on the propagation of information that “Hey, new guest is here, please move to room , please.” This is the first implication.
Ok, one problem solved. But now, there is another implication. As more and more new guests are coming, the receptionist will do the nudge the first room more often creating pulses of sleepy customers politely asking the next door customer to move to the next room. You can imagine that this is going to be a mess.
Finally, do you realise that the first pulse the receptionist make actually is still somewhere at the room asking the room to move? With only one extra guest to fit in, the receptionist practically has created a never ending pulse of unsatisfied customer being disturbed in their slumber. And with number of extra customers to fit in, the receptionist will create an equal amount of customers awake somewhere begging their neighbour to vacate their room and shift to the next room.