The sequence whose limit of difference is zero Let $(x_n)$ be a sequence in $\mathbb{R}^d$ such that 

$$\lim_{n\rightarrow\infty} x_{n+1}-x_n = 0$$

First question: what is the name of such sequence and general properties?

Let $F$ be the set of all adherent values of this sequence. I need to find an example such that F is not connected. How can I think about this question?
Suppose $x_n$ is the partial sum of the harmonic series.
$|x_{n+1} - x_n| = \frac{1}{n+1}$
and $\lim_\limits{n\to\infty} |x_{n+1}-x_n| = 0$, However, $\lim_\limits{n\to\infty} x_n = \infty$
Suppose $x_n$ is the partial sum of the harmonic series.
$|x_{n+1} - x_n| = \frac{1}{n+1}$
and $\lim_\limits{n\to\infty} |x_{n+1}-x_n| = 0$, However, $\lim_\limits{n\to\infty} x_n = \infty$