Let $M^n\subset\mathbb{R}^P$ submanifold, show that there exist a hyperplane $H^{p-1}$ in $\mathbb{R}^P$ such that $H^{p-1}$ intersect $M^n$ tranversally.
In this problem I prove using this: Almost every vector space $V$ with dim$V=l<n$ intersects transversally. I take the set $S\subset (\mathbb{R}^P)^l$ consisting of all linearly independet $l$-tuples of vector in $\mathbb{R}^P$ and prove that $$[(t_{1},...,t_{l}),v_{1},...,v_{l}]\longmapsto t_{1}v_{1}+...+t_{l}v_{l}$$ on $S$ is a submersion. But now I have to prove this without using the previous result and instead of that I have to use Sard's theorem.