Divisions of Numbers in Unit Interval with Numbers in Different Intervals

Theorem
Let $I = \left({0 \,.\,.\, 1}\right)$ be the open unit interval.

Let $a_1, a_2, a_3, \ldots, a_n$ be real numbers chosen in $I$ such that:


 * $a_1$ and $a_2$ are in different halves of $I$


 * $a_1, a_2$ and $a_3$ are in different thirds of $I$


 * $a_1, a_2, a_3$ and $a_4$ are in different quarters of $I$

and so on.

Then $n \le 17$.

That is, for the conditions to be fulfilled, no more than $17$ numbers can be chosen.