Elementary Set Theory, Hilbert's Grand Hotel 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?
 A: Hilbert's hotel is usually brought up as a prelude to a discussion of cardinality, where one of the lessons to be learned is that it is not a sufficient argument to throw up your hands and shout "infinity!" as if that automatically settles everything.
In (a), what you're asked to do is (to stay with the analogy) to construct concrete instructions for each guest: Where should he go, exactly? The instructions should lead to every guest being the only person in a room, and no guest in the rooms that close for renovations. Consider for example

moving each current guest in a room with number $n$ to room number $n + 1$ and lodge the new guest in room 1

That tells everyone exactly where to go, at least after they check the number of the room they're currently in and do a bit of easy arithmetic. What are the similar instructions that will let you shut down half of the rooms?
As for (b), "fully occupied" means there is a guest in each room. That requires an infinite number of guests, but if we're in a universe that has space and building materials for infinitely many hotel rooms, finding people to rent them all out to is a simple matter of marketing.
