Hotel Infinity:  Part Four

The fourth part of Hilbert's Hotel Paradox, on rational numbers and reals.

You run an infinitely large hotel, and are happy to realize that you can accommodate an infinite number of infinite groups of guests.  However, a delicate diplomatic situation arises when a portal to another universe opens, introducing a different kind of guest, in a different kind of group.  Can you make room for them all?  

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