This question was asked at MSe before but with no answer.
I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's fantastic book Topics in Banach space theory.)
I don't see what kind of manipulation with indices $i,j$ is done in order to arrive at the final estimate.
Also,
where exactly we use the assumption that $(x_n)$ is equivalent to the canonical basis of $\ell_1$?
The relevant excerpt:
Look at Giesy's proof, which gives more.
D. P. Giesy, On a convexity condition in normed linear spaces, Trans. Amer. Math. Soc.125 (1966), 114-146.
For every cardinal number (finite or infinite) $N$, if $(x_a)_{a < N}$ is a set of unit vectors that is $K$ equivalent to the unit vector basis of $\ell_1^{N}$, then there are disjointly supported (relative to the basis $(x_a)_{a< N}$ unit vectors $(y_b)_{b < \sqrt{N}}$ that are $\sqrt{K}$ equivalent to the unit vector basis of $\ell_1^{\sqrt{N}}$. The proof is just an exercise. Break the basis $(x_a)_{a < N}$ into $\sqrt{N}$ disjoint subsets each of cardinality $\sqrt{N}$. If no piece "works", use the condition to get a unit vector that witness the non-working--that set of vectors must "work".
