# Question about using Sard's theorem in Whitney's weak embedding theorem

I'm trying to study about Whitney's embedding theorem (the case for $$2k+1$$). Studying the proof of the theorem: http://staff.ustc.edu.cn/~wangzuoq/Courses/18F-Manifolds/Notes/Lec09.pdf I encountered a few things I'm not sure about.

At the end of the third page, it's stated that the image of the map $$\alpha:M\times M\setminus \Delta\rightarrow\Bbb{RP}^{K-1}$$ $$($$where $$\Delta=\{(p,p):p\in M\})$$ is of measure zero following Sard's theorem (because $$M\times M$$ is an open submanifold of dimension $$2k), but Sard's theorem speaks about critical points, so why does it imply that? Same question goes about the map $$\beta$$ mentioned right after it.

for every smooth map $$f: M \to N$$ between two smooth manifolds, let $$X\subset M$$ be the set of critical values $$X = \{ p\in M : df_p : T_pM \to T_{f(p)}N \text{ is not surjective}\}.$$ Then $$f(X)$$ is of measure zero in $$N$$.
If $$\dim M < \dim N$$, then $$df_p$$ is not surjective for all $$p\in M$$ and thus $$X =M$$. Hence the image $$f(M)$$ is of measure zero, according to Sard's theorem.
Remark the statement "If $$\dim M < \dim N$$, then $$f(M)$$ is of measure zero in $$N$$" can be proved fairly easily without the use of Sard's theorem.