TITLE: Convergence of sequences in Hilbert spaces
QUESTION [6 upvotes]: In a Hilbert space, let a sequence $(x_n)$ be weakly convergent to $x$ and be satisfied $\|x_{n+1}-x_n\|\to 0.$ I wonder if we can deduce that $(x_n)$ strongly converges to $x$, or at least it contains a subsequence strongly converges to $x$.
Thank you in advance for your help.

REPLY [5 votes]: This is not true. In $\ell^2$, you can consider the following sequence:
$$
e_1,
\frac12 e_2,
e_2,
\frac12 e_2,
\frac13 e_3,
\frac23 e_3,
e_3,
\frac23 e_3,
\frac13 e_3,
\frac14 e_4,
\frac24 e_4,
\frac34 e_4,
e_4,
\frac34 e_4,
\frac24 e_4,
\frac14 e_4,
\ldots$$
It is easy to check that this sequence has all desired properties.
Now, without a convergent subsequence:
$$
e_1, e_1 + \frac12 e_2,
e_1 + e_2,
\frac12 e_1 + e_2,
e_2,
e_2 + \frac13 e_3,
e_2 + \frac23 e_3,
e_2 + e_3,
\frac23 e_2 + e_3,
\frac13 e_2 + e_3,
e_3,
\ldots
$$