banach spaces - A proof about an unconditional basis theorem

Dan, I doubt that you will find a proof of this in the literature. To get (ii) from (i), note that (i) gives you a block basic sequence $x_1,....,x_m$ in the block subspace $Y$ s.t. $|sum_{i=1}^m x_i|= 1$ and for some choice $a_i$ of signs,

$|sum_{i=1}^m a_i x_i| > C$. WLOG $a_1=-1$. Now group together maximal blocks all of whose signs are

the same to rewrite
$sum_{i=1}^m a_i x_i$ as $sum_{j=1}^n (-1)^j y_j$, where $y_j$ is the sum of the $x_i$'s in the $j$-th maximal block.

I assumed real coefficients in the above, but the complex case is only a bit more involved.

If this explanation isn't sufficient, I suggest that you study further the section in [LT] on bases. I addressed the only point that I thought might be troubling to someone who had studied [LT].