Suppose $\{a_n\}$, $a_n\in \mathbb{R}^n$ is a bound sequence and $\Vert a_{n+1}-a_n\Vert_2\rightarrow 0$. Can we prove that $a_n$ is convergent?
Since $\{a_n\}$ is bound, we know that there are subsequences that are convergent. However, I cannot prove that they converge to a same point.
$\renewcommand{to}{\longrightarrow}$No. Consider $n=1$ and this back-and-forth sequence $$0\stackrel{+\frac12}\to \frac12\stackrel{+\frac12}\to 1\stackrel{-\frac14}\to\frac34\stackrel{-\frac14}\to \frac24\stackrel{-\frac14}\to \frac14\stackrel{-\frac14}\to0\stackrel{+\frac18}\to\frac18\stackrel{+\frac18}\to\frac28\stackrel{+\frac18}\to\frac38\stackrel{+\frac18}\to\frac48\stackrel{+\frac18}\to\cdots$$
This sequence accumulates on the whole interval $[0,1]$, but $\lvert a_m-a_{m-1}\rvert\in O(\alpha/m)$ for some constant $\alpha>0$ (say, $\alpha=100$ should work).