Facebook puzzle solution #3

Oh my, it's been a while. Our third Facebook puzzle(?) was about an infinite hotel and how the idea of infinity can really bend our minds. Here's the puzzle again:

You are the night manager at a hotel with an infinite number of rooms. One night, the hotel is completely full with an infinite number of guests.

But suddenly, a man walks in and rings the bell looking for a room. Can you accommodate him? If so, how?

What if a tour bus arrives with 40 people looking for rooms? Can you accommodate them too?

What if an infinitely big tour bus with infinitely many tourists arrives?

Or to speak of the unspeakable.. What if infinitely many infinitely big tour buses arrive? What can you do to accommodate everybody?

This is called Hilbert's paradox of the Grand Hotel and challenges our concepts of infinity. The hotel is completely full - there aren't any "free spaces" at the very top of the hotel where we can ask guests to move into - yet we can still fit infinitely* many more people in the hotel!

Here's a video from TED-Ed that explains this nicely:

* Countably infinitely