There is a hotel with and infinite number of numbered rooms, each occupied by a single guest. An train with an infinite number of (numbered) coaches, each with an infinite number of (numbered) seats, each occupied by a person, arrives at the hotel. Can you find space for all these people? One method, suggested on Wikipedia, is the triangular number method:
Those already in the hotel will be moved to room $(n^2+n)/2$, or the nth triangular number. Those in a coach will be in room $((c+n)^2+c+n)/2$, or the $(c+n-1)$ triangular number, plus $(c+n)$. In this way all the rooms will be filled by one, and only one, guest.
So the person who's in room $1$, stays in room $1$. The person in room $2$, moves to room $3$. The person in room $3$ moves to room $6$, and so on.
What about the new guests? If they are moved to room $((c+n)^2+c+n)/2$, can't that still be a triangular number and hence already occupied?