Suppose a bounded sequence $(x\_n)$ converges to $x$ in the Cesaro sense (i.e., $\frac{1}{n}(x\_1 + x\_2 + \dots + x\_n)\rightarrow x$) in a separable Hilbert space $H$. How to prove that some subsequence $(x\_{n\_k})$ converges weakly to $x$?



Cesaro convergence implies weak convergence of a subsequence
If we take $x\_n = (-1)^n x$ then $x\_n$ converges to $0$ in Cesaro sence. But no subsequence of $x\_n$ converges weakly to $0$. $x\_n$ is also a bounded sequence.
Hence your statements seems wrong.


