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I've seen the paradox mentioned a lot but I don't see it as paradox at all.
A hotel with infinite rooms has infinite size.
Just idea of asking guests to move would take infinite time (i.e will never happen).
if each guest wait for the his new room to vacate, no one would move at all.
infinite guests with infinite different leave times. so one room most be free.
If they all step out and into the new rooms at the same time, it's possible.
It's not a mathematical paradox, only against intuition.
But how would they all know when to step out and into new room?
They need some kind of a signal (bell, phone call, flashing light, speaker instructing them to move, ...)
Fastest of these communication is light which has maximum speed of 300,000 km/s
The light will reach first room instantly but since number of rooms is infinite, the light need infinite time to reach that last room
It's an illustration of infinities, not an actual hotel.