[3] After returning to Vienna, Helly worked as a tutor, gymnasium teacher, and textbook editor until World War I, when he enlisted in the Austrian army.
[5] After a complicated return trip, Helly finally came back to Vienna in 1920, married his wife (mathematician Elise Bloch) in 1921, and also in 1921 earned his habilitation.
Unable to obtain a paid position at the university because he was seen as too old and too Jewish, he worked at a bank until the financial collapse of 1929, and then for an insurance company.
[7] Helly's proof only covers continuous functions over closed intervals of the real numbers; the more general theorem requires the ultrafilter lemma, a weakened variant of the axiom of choice, which had not yet been invented.
[8] His most famous result, Helly's theorem on the intersection patterns of convex sets in Euclidean spaces, was published in 1923.