I have this one other question and I think it requires a rigorous proof involving balls etc and I have no idea about how to go about it.
What do you know about the boundary?
For instance, do you know that Boundary(S) = Closure(S)\Interior(S)?
If you do, then
Boundary(open solid unit ball) = (the closed solid unit ball)\(the open solid unit ball) = the unit ball.
Edit: Ah, you wanted a points-and-balls proof. I'll let this be anyway.