Set of Subsequential limits of a sequence is connected

Let $x_n$ be a bounded sequence in $\mathbb{R}^n$ such that $\sum_{k\in \mathbb{N}} \| x_{k+1} - x_{k}\|^2 < +\infty$.

Let $S$ be the set of all subsequential limits of $x_n$, that is:$$S = \{x~|~\exists \text{a subsequence of }x_{r_n} \text{ of } x_n\text{ such that } x_{r_n}\rightarrow x\}$$

Then show that $S$ is connected.

I have tried to show by contradiction that if $S= U\cup V$ such that $U,V$ are open and $U\cap V=\phi$, then the since the sequence is bounded (all subsequential limits are finite), the sequence itself can be broken into two parts that lies eventually in one of $U$ or $V$. Then I think I need to use the fact $\lim \| x_{k+1} - x _k \| = 0$ to get to a contradiction. However I am unable to do so.