Answer
We move the current guests to the odd-numbered rooms.
The new guests receive the even numbered rooms.
Work Step by Step
Since Hilbert's Grand Hotel has a countably infinite number of rooms, we
can number the rooms by positive integers $Z^+: 1,2,3,4,...$
The hotel is fully occupied and a countably infinite number of guests arrive.
We move the current guests to the odd-numbered rooms. This then means that the guest in room $1$ stays in room $1$,guest in room $2$ moves to room $3$, guest in room $3$ moves to room $5$, guest in room $4$ moves to room $7$.
The new guests receive the even numbered rooms. The first new guest thus
receives room $2$, the second new guest receives room $4$, the third new guest receives room $6$, etc
We then note that all current guests will still have a room and all new guests will also receive a room, thus no current guests had to be evicted.
