[mistercow]

No, the problem was phrased correctly. What we are saying is that for each natural number n, room n is occupied by a guest. This is the mathematical idea embodied by "an infinite number of occupied rooms". If you say "an infinite number of guests and an infinite number of rooms", then the question becomes obvious and trivial.

What makes it interesting is that we really do mean that every room is occupied to begin with. The surprising result is that we can in fact add another guest by having everyone move over one room, even though we started out with every room occupied. And in fact, we can add infinitely many new guests by telling everyone to move to their room number \* 2.