Divisions of Numbers in Unit Interval with Numbers in Different Intervals

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Let $I = \openint 0 1$ 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.