How to prove the convergence for function of function series?
Say, here're two examples
- Given $x_1>0, x_{n+1}=\ln(1+x_n)$, Prove $\lim_{n\to\infty}nx_n=2$
- Given $0
, Prove $\lim_{n\to\infty}\sqrt{n}x_n$ exist, and give this limit.
I've written programs to check the above two problems, and it seems the assertions are true, however I found proving this kind of problems extreamly hard, since simply expanding this function of function series usually make me totally lost. Any one can give me some suggestions on such problems? Thanks a lot!