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

Question

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 $2k<K-1$), 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.

Answer 1

The Sard's theorem says that

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.