I just encountered the following statement: *any subsymmetric basic sequence is either weakly null or equivalent to the unit vector basis of $\ell\_1$.*


It's the first time I realize that. I do see the case in which it is equivalent to the unit vector basis of $\ell\_1$. However, I cannot prove that if that's not the case, it must be weakly null. Is this "trivial"? Also, while thinking about that, I came up with a couple of questions: is every basic sequence bounded away from zero? What about a subsymmetric basic sequence? So far I cannot come up with a counterexample.


I'm not looking for a complete answer. A hint or reference would suffice.



Classification of subsymmetric basic sequences
In the theorem you quote, part of the definition of subsymmetric includes the hypothesis that the sequence is unconditionally basic. If you do not include this in the definition of subsymmetric, then the summing basis for $c\_0$ is an example of a weakly Cauchy subsymmetric basis that is not equivalent to the unit vector basis of $\ell\_1$.


