You're making very strong assumptions about the number of people there are. How did you decide that's a countable infinity? No such assumption is made in the

You're making very strong assumptions about the number of people there are. How did you decide that's a countable infinity? No such assumption is made in the problem statement. If there are an uncountably infinite number of people then they wouldn't all fit in the countably infinite number of rooms, so of course there are people that can show up.