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    It’s too bad that this cool trick isn’t constructive.

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      I liked the writeup of the problem and the proof quite a bit. I think the $f(x) = x_a$ definition, while it makes sense in context, is a pretty awful abuse of notation, because the author implicitly went from choosing any basis vector a from the Hamel base into fixing a specific a. I think mentioning something like Now fix an $a \in \mathbb{H}$ would be helpful.