Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to $(e_n)$. I would like to claim that if furthermore $(x_n)$ is normalized (not just seminormalized) then we can choose the subsequence such that the equivalence constant is bounded by some absolute constant.
In other words, I claim the following: There exists an absolute constant $C\geq 1$ such that if $(x_n)\subset\ell_p$ is normalized and weakly null then there exists a subsequence $(x_{n_k})$ which is $K$-equivalent, $1\leq K\leq C$, to $(e_n)$.
I am looking at the proof to the nonuniform version in Topics In Banach Space Theory by Albiac/Kalton, and it looks at first glance like my claim is true. In particular, looking at the proofs to Theorems 1.3.9 and 1.3.10, which are used in Albiac/Kalton to prove equivalence, it looks like we can get the equivalence constant to be bounded by 17 (the book appears to use $\theta=8/9$, where the equivalence constant is bounded by $\frac{1+\theta}{1-\theta}$). However, I have not worked through the details carefully, so maybe I am wrong.
Is there a reference to this claim? Eventually, if this claim is true then I will need to mention it in a paper I'm writing. In that case, it would be silly to have to prove it from scratch since that would mean re-hashing the well-known proof of the nonuniform result found in Albiac/Kalton and countless other books and papers. Far better to just have a reference.
Thanks guys!