The proof I'm referring to can be found here.1
I have worked through this proof and am literally at the last line of the UBP proof but am stuck. I have two questions:
Firstly, how does he obtain the inequality $\| x - x_n \| \leq \frac{1}{2}3^{-n}$? I can get $\| x - x_n \| \leq (6)3^{-n}$ via geometric series (unless I've done the arithmetic wrong!). I suspect this will do, based on what follows, but I'd still like to know how to get this estimate.
Secondly, immediately after this, he has an inequality $\| T_n x \| \geq \frac{1}{6} 3^{-n} \| T_n \|$. Where does this inequality come from? Seems like I'm missing something simple here again.
In any case, thanks in advance!
1Sokal, Alan D., A really simple elementary proof of the uniform boundedness theorem, Am. Math. Mon. 118, No. 5, 450-452 (2011). ZBL1223.46022, MR2805031.