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in this theorem of the paper "Uniformly Non-Square Banach Spaces" of Robert C. James:

Theorem: A Banach space is refiexive if its unit ball is uniformly non-square.

why $K _{n}$ is positive, monotone increasing and for any number $r$ with $1> r > 1-\delta $, there is a positive number $\epsilon $ and an $N$ such that

\begin{equation*} \frac{K _{n} - \epsilon}{K _{n} + 2\epsilon} > r > 1 - \delta \end{equation*}?

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Let $\epsilon =\frac {(1-r)K_1}{6}.$ For all $n$ we have $$0<1-\frac {K_n-\epsilon}{K_n+2\epsilon}=\frac {3\epsilon}{K_n+2\epsilon}<\frac {3\epsilon}{K_n}\leq \frac {3\epsilon}{K_1}=\frac {1-r}{2}<1-r<\delta .$$

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  • $\begingroup$ Thanks for helping :) $\endgroup$ – G. P Feb 5 '18 at 14:45

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