TITLE: Is it possible for $x_{n+1}-x_{n}$ to converge to $0$, but the limit of $x_n$ to not exist?
QUESTION [5 upvotes]: I was reading about subsequences today and I came across an example which said the if $x_{n+1}-x_{n}$ converges to $0$, it's possible for $x_n$ to not have a finite limit (for example if we take $x_n$ to be the harmonic series). But then I thought "is it possible for the limit of $x_n$ to not exist at all?" and I think the answer is no,because I couldn't find any examples, but I am not sure. Could you please tell me what your opinion is on this?
Thanks in advance!

REPLY [3 votes]: Let
$$x_n-x_{n-1}=\frac{(-1)^{\lceil\log_2n\rceil}}n$$
This is in effect taking the harmonic $x_n=H_n$ example, which has $\infty$ as a limit, and "folding it" into infinitely many intervals where $x_n$ goes up and down alternately by at least $\frac12$ per interval:

From $x_2$ to $x_4$, $\frac13+\frac14>\frac12$ is added
From $x_4$ to $x_8$, $\frac15+\frac16+\frac17+\frac18>\frac12$ is subtracted, etc.

Thus $x_n$ cannot have a limit.