  Consider a subset of positive integers $S$. In this paper, we reduce the
upper bound on the length of a minimum program that enumerates $S$ in terms of
the probability of $S$ being enumerated by a random program.
  So far, the best-known upper bound was given by Solovay. Solovay proved that
the minimum length of a program enumerating $S$ is bounded by $3$ times minus
binary logarithm of the probability that a random program enumerates $S$.
Later, Vereshchagin showed that the constant can be improved from $3$ to $2$
for finite sets. By improving the method proposed by Solovay, we demonstrate
that any bound for finite sets implies the same bound for infinite sets, modulo
logarithmic factors. Thus, the constant can be replaced by $2$ for every set
$S$ due to the result of Vereshchagin.
