I am reading Elements of Information Theory (Thomas M. Cover, Joy A. Thomas, 2nd edition) in which the following theorems are given (page 475--476):
- For any integer $n$
$$ K(n) \leq \log^* n + c. $$
- There are an infinite number of integers such that $K(n) > \log n$.
Assuming both statements true, this implis that there are an infinite number of integers $n$ such that
$$ \log n < \log^* n + c. $$
Knowing the growing rate of $\log$ and $\log^*$ this does not seem possible if $c$ is independent of $n$... Is it? If not, which one of the two above statements is false?