sequences and series - $a_{n+1}=log(1+a_n),~a_1>0$. Then find $lim_{n rightarrow infty} n cdot a_n$

Suppose that $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$.

I can find $\lim_{n \rightarrow \infty}a_n=0$. But I have no idea how to find $\lim_{n \rightarrow \infty} n\cdot a_n$. (In this problem, $\log$ denotes the natural logarithm.)
Could you give me key idea about solving this problem?