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Is the existence of Banach limits independent of ZF+DC?

Assuming this is known, where can I find a proof?

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    $\begingroup$ See Theorem 44 here $\endgroup$
    – Wojowu
    Commented Mar 2, 2022 at 23:02
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    $\begingroup$ @Wojowu. Thank you. $\endgroup$
    – Philip Ehrlich
    Commented Mar 2, 2022 at 23:33
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    $\begingroup$ @Wojowu: I was going to say that! :-) $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 3, 2022 at 0:16
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    $\begingroup$ @PhilipEhrlich: Hmm. I can't really remember, to be fair. This result will either appear directly in Eric S. Schechter's book, probably with a proper reference, or it will be easily found through Google Scholar or something like that. If I had to guess, I'd say either Banach himself noted something like that to be true or maybe it was in one of the many works of Pincus and Solovay on these topics. But I wouldn't be shocked to learn that Kuratowski or Sierpinski were the actual observers. Martin Väth hangs around here sometimes, maybe he'll know. $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 3, 2022 at 0:55
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    $\begingroup$ @Asaf.Thank you. The proof you offer is lovely. $\endgroup$
    – Philip Ehrlich
    Commented Mar 3, 2022 at 1:00

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