Question 61
Let $S$ be the rational numbers in $[0,1]$. An infinite real sequence $\{x_1,x_2,\ldots,x_k,\ldots\}$ is called “beyond imagination” if there is a function $H:S\to\mathbb R$ satisfying:
(i) $H$ is nondecreasing on $[0,1/2]$: for rational $0\le a\le b\le1/2$, $H(a)\le H(b)$;
(ii) for integers $p\gt 0$ and $0\le q\le p$,
$$H(q/p)=\frac{\sum_{k=1}^{q}x_{p+1-k}-\sum_{k=1}^{q}x_k}{p}.$$
Find all sequences beyond imagination.
(i) $H$ is nondecreasing on $[0,1/2]$: for rational $0\le a\le b\le1/2$, $H(a)\le H(b)$;
(ii) for integers $p\gt 0$ and $0\le q\le p$,
$$H(q/p)=\frac{\sum_{k=1}^{q}x_{p+1-k}-\sum_{k=1}^{q}x_k}{p}.$$
Find all sequences beyond imagination.