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A clever proof of Banach-Steinhaus theorem (uniform boundedness principle): Alan D. Sokal, A really simple elementary proof of the uniform boundedness theorem, Amer. Math. Monthly 118, 450-452 (2011); arxiv.org/abs/1005.1585 dx.doi.org/10.4169/amer.math.monthly.118.05.450
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See here: Alan D. Sokal, A really simple elementary proof of the uniform boundedness theorem, Amer. Math. Monthly 118, 450-452 (2011) https://arxiv.org/abs/1005.1585, http://dx.doi.org/10.4169/amer.math.monthly.118.05.450
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The proof I'm referring to can be found here. 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 ...
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