Does there exists a "simple" proof that every smooth map $f: M\to N$ between manifolds contains at least one regular value, without using Sard's theorem?
Motivation: In Milnor, Brouwer's fixed point theorem is proved by means of Sard's theorem, but it is only used for the claim that a map from an $(n+1)$-disc into the boundary sphere contains at least one regular point (if I understand the text correctly). This seems much weaker than Sard's theorem that states that almost all values are regular.