Uncountability of Hamel Basis for Banach Space

Last week, I was discussing with a student some elementary details of normed and Banach spaces. The subject of Hamel bases came up, and I incorrectly said that a Hamel basis for an infinite-dimensional normed space is necessarily uncountable. After thinking about the proof I had in mind, I realized that I needed the additional hypothesis of completeness. Here’s my proof, if you’re interested.

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