In my earlier proof of the uncountability of a Hamel basis for a real or complex Banach space, I had knowledge of a slightly non-trivial result in introductory functional analysis–namely, that any finite-dimensional subspace of a real or complex Banach space is closed with respect to the norm topology. I have updated my earlier solution to include a fairly complete sketch of the proof of this “lemma.” The new version can be found here.

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