Is there any visual proof that rationals in cantor set are dense in cantor set?

Question

I was thinking about how we can get a feel that rationals in Cantor set are dense in Cantor set. Is there any way to put this thing in a visual way? It is quite easy to think for irrationals in the cantor set. Because if we take any open-ball in Cantor set, then it is basically an interval intersection the Cantor set and of course non-empty, but any point of cantor set in particular that in the interval is a condensation point of cantor set, there are countably many rationals, so others are irrationals of cantor set. So, the open-ball in cantor set intersects irrationals in cantor set. But for rational, I need a visual argument. One way to understand I think is taking an open ball in cantor set. Since, this interval contained some fragments that have been removed. So, there will be those endpoints of segments in that interval. The endpoints are always rational. So, this interval contains rationals. But I am looking for a visual approach that would be easier and intuitive.

Answer 1

The boundary points of the Cantor set (i.e. the boundary points of the removes intervals) are rational and dense in the Cantor set.