"Picture a hotel with rooms going on forever, and every single room is occupied. Sounds full. But if one new guest arrives, the manager just moves everyone from room N to room N plus 1, opening the first room. Weirdly, full still has space. Bring 100 more guests, same trick. Shift everyone forward by 100. Now bring an infinite bus. Still works. Move every current guest into even numbered rooms, and all the odd rooms open up for the newcomers. Even infinitely many buses with infinitely many passengers can fit, if you can list every passenger in a countable order. That is the key word, countable. If something can be matched one-to-one with one, two, three, and so on, the hotel can handle it. But here's the twist. Some infinities are bigger. The real numbers cannot be listed that way. They are uncountable. So yes, an infinite hotel can run out of room, just not for the reason you expect."