TITLE: On a certain sequence
QUESTION [5 upvotes]: Let $\{y_n\}$ be a sequence defined by $y_n=x_n-\ln n$ , where $\{x_n\}$ is defined by $x_1=1 ; x_{n+1}=x_n + e^{-x_n} , \forall n\in \mathbb N$ . Then is $\{y_n\} $ bounded ? Is $\{y_n\}$ convergent ? I know that $\{x_n\}$ is strictly increasing and unbounded ; but I am unable to say anything about $\{y_n\}$ . Please help  .Thanks in advance

REPLY [3 votes]: It is clear that the increasing sequence $x_n \to +\infty$ Put $z_n=\exp(x_n)$, $z_n\to +\infty$ also. We have that $\frac{z_{n+1}-z_n}{z_n}=\exp(1/z_n)-1\sim \frac{1}{z_n}$, this show that $u_n=z_{n+1}-z_n\to 1$. Using Cesaro's theorem, we get that $\frac{u_1+\cdots+u_n}{n} \to 1$, hence $z_n/n\to 1$, and $\log z_n-\log n=x_n-\log n\to 0$.