In Milnor's lectures on H-cobordism theorem, there is a theorem without a proof. There are references, but I don't have access to them.
Theorem: Let $f \colon M \to N$ be a smooth map which is an embedding on a closed subset $A \subseteq M$. Provided $\dim N > 2 \dim M$ there exists a smooth embedding $g \colon M \to N$ homotopic to $f$ and such that $g|_A = f|_A$.
Is it possible to prove this theorem "by hand", i.e. without function spaces?