There is no such thing as infinity +1 because infinity is not a number.
The hotel thing works;
Given that the people follow the owner's instructions in regards to changing rooms, there will never be a room conflict, and everybody will indeed fit.
The person in room 1 moves to room 2. There is nobody in room 2 because that person moved to room 4. Person 2 was able to move into room 4 because *that* person moved into room 8.
The person in room 3 moves to room 6. Person 6 moved to room 12, and 12 to 24.
Everybody leaves their room. And each person is given a unique new room number to move into. One person per room, and nobody left without a room. Therefore everybody fits.
Infinity is funny that way
The set of positive, even integers (the original guests in their new rooms) is the same size as the set of all positive numbers (the original guests ... and also the original guests in new rooms plus the frat boys)