The following two questions are on an assignment of mine:
a) Suppose that Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel.
b) Show that a countably infinite number of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
All the information I've been told about Hilbert's Grand Hotel is:
1) Hilbert’s Grand Hotel is a paradox. The Grand Hotel has a countably infinite number of rooms, each occupied by a guest. A new guest is accommodated by moving each current guest in a room with number n to room number n + 1 and lodge the new guest in room 1.
I am confused by the question and what exactly it is asking. My understanding of the Hotel is that, the rooms are functionally infinite. Therefore, don't both of the questions answer themselves?
Even if you close half of all the rooms, theres still an infinite number of them, so you will not need to send any guests away.
Because theres an infinite number of rooms, how can the hotel ever be considered fully occupied? Wont there always be a room n+1?