There is this
<a href="https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#School_of_Names"
  >old paradox</a
>
about moving halfway to a point, then halfway again, and so on. I'm used to
hearing it referred to as one of of Zeno's paradoxes, but evidently there's
documents from Hui Shi who separately thought about this around the same time as
Zeno.
<br /><br />We can use this paradox to squeeze all the numbers into a box!
First, draw a line; second, place a number at the center of your line; third,
place the successive number halfway to the end of the line; then place the next
number halfway from the previous number to the end of the line; repeat. Reverse
this process to get the numbers going off to the other side of the line but with
decreasing numbers instead.<br /><br /><br /><br />
This process is directly related to
<a href="https://en.wikipedia.org/wiki/Geometric_series">geometric series</a>.
<a href="https://activecalculus.org/single/sec-8-2-geometric.html#Zaw"
  >Here's a reference</a
>
on how to do that process in a smooth way to enable smooth dragging. This can
help with the construction of a bijection between an open interval and the real
numbers.