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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?

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    $\begingroup$ This is just a general position argument; the proof is exactly the same as in the standard Whitney embedding. $\endgroup$ Mar 19, 2018 at 14:38

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