$u_{n+1}=\log\left(\dfrac{e^{u_n}-1}{u_n}\right)\,$ with $\,u_0=1.$
I can see by computing the terms that the sequence is monotonic decreasing and going to $0$. However, I can’t figure out how to prove this.
I think I have proven boundedness by induction.
Assuming $\,u_n\,$ is less than $1$, I proved that $\log\left(\dfrac{e^{u_n}-1}{u_n}\right)<1\,$ as doing some arrangements I ended up with $\,u_n<\log(u_n)+\log\left(e+\dfrac1{u_n}\right)\,.$
As the second logarithm is greater than $1$ for every $\,u_n\,$ positive (it is bounded below by $0$ at it is a logarithmic function) I ended up proving the assumption.
Please correct me if I am mistaken.
How do I find a limit now?