Let $X$ be a Banach space with basis $(e\_n)\_{n=1}^\infty$, and suppose that $(x\_i)\_{i=1}^\infty$ is a normalized block basic sequence of $(e\_n)\_{n=1}^\infty$. In addition assume that $(x\_i)\_{i=1}^\infty$ is subsymmetric, weakly null, and such that $\sup\_i \left\vert \text{supp}(x\_i) \right\vert = \infty$.


My question: is it possible to extract a subsequence $(x\_{n\_j})\_{j=1}^\infty$ for which $\sup\_j \left\vert \text{supp}(x\_{n\_j}) \right\vert < \infty$? Perhaps find an equivalent normalized block basic sequence $(y\_j)\_{j=1}^\infty$ of $(e\_n)\_{n=1}^\infty$ for which $\sup\_j \left\vert \text{supp}(y\_j) \right\vert < \infty$?



Extracting a subsequence for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$
No, take, for instance, a reflexive Orlicz sequence space $\ell\_M$ not isomorphic to $\ell\_p$ and a block sequence $(x\_i)$ equivalent to $\ell\_p$ basis that space contains. Every block sequence in the space with uniform bound on the support will be equivalent to the unit vector basis of $\ell\_M$. 


