How to find the limit of these sequences? Let $\{a_n\}$ be a real-valued sequence such that $a_1 \geq 0$ and $$a_{n+1}=\ln(a_{n}+1)$$ for all $n\ge1$. How can we find the following limits?
$$\lim_{n\to \infty}na_n=?,$$ $$\lim_{n\to \infty}\frac{n(na_n-2)}{\ln n}=?$$
Thanks in advance.

A physisist's approach to the 1st limit (not proof at all): first of all it is clear that $a_n\rightarrow 0$ as $n\rightarrow\infty$. The question is - how? Supose that $a_n=a(n)$ can be modeled by a smooth function of $x$. Then, expanding the left side and right side of the recursion relation into Taylor series to "first order", we find the ODE
$$a'(x)=-\frac12a(x)^2$$ 
with the obvious solution $a(x)=\frac{2}{x}$. So a physicist would conclude that  $\lim_{n\rightarrow\infty}na_n=2$.
