I have the following question, that a friend of mine asked me yesterday:
Let $H$ be a Hilbert space, with norm $\|\cdot\|$. Let $(x_k)_{k \ge 0}$ be some sequence in $H$. Assume that $x_k$ is bounded (so that $\|x_k\| \le C$ for all $k$), and that $(x_k)$ has bounded L2 variance, i.e., $\sum_{k \ge 0} \|x_{k+1}-x_k\|^2 < +\infty.$
Does it then follow that the sequence $(x_k)$ converges?
If not for all Hilbert spaces, then
Does it at least follow that $(x_k)$ converges for $H=\mathbb{R}^n$?