Hilbert Hotel

Let a hotel have a Denumerable set of rooms numbered 1, 2, 3, .... Then any finite number $n$ of guests can be accommodated without evicting the current guests by moving the current guests from room $i$ to room $i+n$. Furthermore, a Denumerable number of guests can be similarly accommodated by moving the existing guests from $i$ to $2i$, freeing up a Denumerable number of rooms $2i-1$.

References

Lauwerier, H. ``Hilbert Hotel.'' In Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, p. 22, 1991.

Pappas, T. ``Hotel Infinity.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 37, 1989.