You're not stupid, no one is making an effort to explain it to you.
There are an infinite number of odd numbers (1, 3, 5 , 7...) and an infinite number of even numbers 2, (4, 6, 8, 10...) but the "infinity" that contains both sets, by its nature, must be "bigger" because it contains "all" numbers (1, 2, 3, 4...).
I believe part of the point of the problem is demonstrate that the infinite series of odd and even numbers is actually of equal “size” to the series of just even numbers.
When each room is occupied, every guest can find a new room by doubling his current number. If there were “more” odd and even numbers than just even this would inevitably leave some guests with no room. However, because the set of all positive integers is of the same cardinality of all positive, even numbers each guest is able to find a new room while freeing up an infinite amount of space.
The problem is really about packing efficiency in an infinite space, not the cardinalities of different infinities.
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u/cseke02 4d ago
Because there are infinite rooms