If there were one person for every natural number, then you could put every even-numbered person in room n/2 (where n is the number for that person) and still f

If there were one person for every natural number, then you could put every even-numbered person in room n/2 (where n is the number for that person) and still fill up the entire hotel. This is because even numbers are infinite, and you can find a one-to-one relation with natural numbers (just divide the even number by 2). In that case, the infinite hotel is filled, and you still have an infinite amount of people outside it that you can accommodate (the odd-numbered ones) :)