The Lost Room in The Grand Hotel

David Hilbert suggested that we:
consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms... (Wikipedia)

That's the paradox, the hotel can always accommodate one more even though every room is occupied. Infinity has struck again.