Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."):


"Assume $(x\_n)$ converges weakly to $x$. Then there exists a sequence $(y\_n)$ made up of convex combinations of the $x\_n$'s that converges strongly to $x$.''


The Lemma says that "there exists a sequence...''.


Is it true that $\textbf{every}$ sequence $(y\_n)$ made up of convex combinations of the $x\_n$'s converges strongly to $x$?


In particular, if we consider 
$$y\_n = \frac{1}{n}(x\_1 + x\_2 + ... +x\_n),$$


is it true that $y\_n$ converges strongly to $x$?



A question involving Mazur's Lemma
If $x\_i= u\_{floor(log(i)+1)}$ where $\{u\_i| i=1,2,\ldots\}$ is an orthonormal basis for a Hilbert Space, then $(x\_i)$ converges weakly to $0$ and $y\_n = 1/n \sum\_{i=1}^n x\_i$ does not converge strongly to anything. 


In particular, if $n = floor( e^k )$ for an integer $k\geq1$, then $||y\_n - u\_k|| < 2/3$ hence $||y\_n||>1/3$.


Interestingly, if we define the linear operator $L\_n: l\_2\rightarrow R$ by


$L\_n(x) = 1/n \sum\_{i=1}^n x(i)$


then the $L\_n$ converge arbitrarily slowly to the zero operator. 


