Since I can't (yet) leave a comment to Andres Calceido's answer.
A link to another paper by Luxemburg with a proof of the Hahn-Banach Theorem using ultrapowers http://www.ams.org/bull/1962-68-04/S0002-9904-1962-10824-6/S0002-9904-1962-10824-6.pdfhttps://www.ams.org/bull/1962-68-04/S0002-9904-1962-10824-6/S0002-9904-1962-10824-6.pdf