Suppose I have a closed, bounded, non-degenerate interval $\displaystyle I = [a, b]$.

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 $\displaystyle I_{1}=[a, c]$ and $\displaystyle I_{2}=[c, b]$.

My next action is to find the midpoints of $\displaystyle I_{1}$ and $\displaystyle I_{2}$, lets call them $\displaystyle c_{1}, c_{2}$ 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?