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?