Find distinct positive real numbers $x\_1$ , $x\_2$ , ... of least supremum such that, for each positive integer $n$, any two of 0, $x\_1$ , $x\_2$ ,..., $x\_n$ differ by $1/n$ or more.


Note that the hurdle term $1/n$ is optimal in the sense that any replacement for it would need to stay below a constant multiple of it to allow a finite supremum. By a nonconstructive proof, there is a unique solution minimal with respect to the lexicographic ordering of real sequences; so a constructed solution (while eluding me) doesn't seem impossible.
 Although I haven't seen this problem anywhere, it looks too simple not to have been posed before. Any pointers would be welcome. 



A sequential optimizing task
Belated thanks to Kevin O'Bryant for his pointer to discrepancy theory. This led me eventually to a source where the problem is solved: See Theorem 6.7 in Harald Niederreiter's book *Random Number Generation and Quasi Monte Carlo Methods* (SIAM 1992). The logarithmic sequence described by Tracy Hall is due to Rusza and is indeed optimal. 


