Multiply all the digits of a number $n$ by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative persistence of $n$.
According to http://mathworld.wolfram.com/MultiplicativePersistence.html, it was shown by Erdos that:
Ignoring all zeros, at most $c\ln(\ln n)$ steps are needed to reduce $n$ to a single digit, where $c$ depends on the base.
Where can I find a proof of this result?