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 $\ell_1$?
