Aren't there already guests in room n+1?
I'm no mathematician, but this seems to be an edge phenomenon that everyone is willing to ignore. I suppose you could just as easily tell the new guests to run down the hallway until they find an empty room at the end; out of sight, out of mind...
There is no room 'n+infinity'. There's just not such a room, anywhere, even in principle. Each guest n can move to room n+1 because the guest there is moving to n+2. This works. That's the point.
And you couldn't, in fact, tell the guests to run down the hallway to find an empty room: There is no empty room until you move everyone.
Infinity is counterintuitive, you say? Well, yes, that's the point of this illustration.
You are certainly correct, however, that Cantor's hotel does not need to have any rooms with transfinite room numbers.
That is, unless I am also thoroughly confused.
You are, but that's OK; they call it a "veridical paradox" for a reason.
There is a mapping of every guest to a room: guest n is in room n. Yet somehow there exists a room that has no guest, despite being able to model both rooms and guests with the same infinite set.
Incidentally, this is one way to define infinite sets. A set is infinite if and only if there exists a proper subset that has the same size (cardinality).
What makes it interesting is that we really do mean that every room is occupied to begin with. The surprising result is that we can in fact add another guest by having everyone move over one room, even though we started out with every room occupied. And in fact, we can add infinitely many new guests by telling everyone to move to their room number \* 2.
For example. All blocks are either red or blue. You have an infinite number of blocks that are all red. Do you have any blue blocks? No. Where is the mistake in my logic?
1. Have all guests of the hotel simultaneously pack their things and move out of their room onto the hallway.
2. Have all guests simultaneously move along the hallway to be standing in front of the room next door (in the same direction, obviously).
3. Have all guests simultaneously move into the room they are now standing in front of.
The first room is now empty, with nobody having had to leave the hotel, and every guest still has his/her own room.
No, because the former occupants of n+2 are now in n+3.
Because every natural number n has a successor n+1, you can do it for all of them.
Each n except 1 has a predecessor n-1 from which guests where moved to n. Nobody moved into 1, therefore you can put the new arrivals there.
After that, there are still (countably) infinite many guests. They are just matched up differently with the rooms/numbers.
Because you have infinitely many rooms, you were able to accomodate one more by matching them up differently. That's sort of the point :-)
Ponder this if you will: how did the hotel get full in the first place? if you assume that is possible, then you can reverse the original room assignments, and reassign rooms.
The source of your confusion is that you trusted the problem is even possible to set up, without availing yourself of the power that the setup implies.