Answer
Move each guest from room $n$ to $2n-1$
Work Step by Step
Since Hilbert's Grand Hotel has a countable infinite number of rooms, we
can number the rooms by positive integers $Z^+: 1,2,3,4.....$
The even numbered rooms are closed down
We can then move each guest from room $n$ to room $2n -1$
$f: Z^+ \implies Z^+, f (n) = 2n- 1$
This then means that the guest in room $1$ stays in room $1=2(1)- 1)$, guest in room $2$ moves to room $3(= 2(2) -1)$, guest in room $3$ moves to room $5( =2(3)-1)$, etc
We then note that each guest then still has a room and thus all guests can
remain in the hotel
