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As I have learnt in my lecture, the proof to Sard's theorem at the last step has shown if $k>\frac{n}{m}-1$ the $f(C_k)$ has zero measure in $M$. But the statement of Sard's theorem said $k\geqslant n-m+1$. $C_k$ represents the set of all derivatives till $k$-th order are vanishing.

How does it come?

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    What is $C_k$? Please explain your notation. – 2017-01-18
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    this proof: http://web.stanford.edu/~amwright/Sard.pdf – 2017-01-18

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