Suppose I have a closed, bounded, non-degenerate interval .
I have a set X initially containing the points a and b.
My first action is to find the midpoint of I, call it c, and add c to X.
I now have two new intervals and .
My next action is to find the midpoints of and , lets call them and add them to X.
I now have four intervals.... and I continue this procedure until I can subdivide no more.
Does X contain all points in the original interval I? Or stated differently, is there a point in I that is not in X?