I find your proof very hard to follow. That is not saying it is incorrect!
I think it is too complicated.
It is easy to prove that any open set is simply the union of balls.
The interior is just the union of balls in it.
The complement of the closure is just the union of balls in it.
The complement of the boundary is just the union of balls in it.
To follow that last bit, think this way.
If the there is a ball that is a subset of or the complement of
That is it contains only points of or points not in .