Interesting use of relations in calculus.

I don't understand the transition from here to the next statement below. It is strange that you abandon that you have chosen, and in the following appears only as a bound variable.

To have one has to show that there are points in that are arbitrarily close to , i.e., that does not, for example, begin with end end with . That, I think, is the main thing that one needs to show.

I think this lemma is valid. Since for each point we have a neighborhood, you can choose a finite covering of . Then, e.g., by induction on the number of neighborhoods in this covering it is easy to show that .