... modified1
As pointed out by M. Hutter (14 Nov. 2000, personal communication) and even earlier by A. Fujiwara (1998, according to P. M. B. Vitányi, personal communication, 21 Nov. 2000), the proof on the bottom of p. 249 of [30] should be modified. For instance, the sum could be taken over $x_{i-1}<xleq x_i$. The sequence of inequalities $sum_{x_{i-1}<xleq x_i}P(x)>x_iP(x_i)$ is then satisfiable by a suitable $x_i$ sequence, since $liminf_{xtoinfty}\{xP(x)\}=0$. The basic idea of the proof is correct, though, and quite useful.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.