Let $M\subseteq \mathbb{R}^p$ an n-dimensional submanifold. Show that there is a hyperplane in $\mathbb{R}^p$ that intersects $M$ transversally.
My ideas:
I shall use the theorem of Sard but I don't really know how to apply it. I can write the hyperplane as $H^{p-1}=f^{-1}(c)$ where $f:\mathbb{R}^p\to\mathbb{R},\, f(x_1,...,x_p)=x_k$ with $k\in\{1,...,p\}$. Also what I have to show is that $T_xM\oplus T_xH^{p-1}=\mathbb{R}^{p}$ for all $x\in M\cap H^{p-1}$. I know that $T_x H^{p-1}=H^{p-1}$ and therefore I must "only" find $H^{p-1}$ such that it intersects $M$ at a point or points where $T_x M$ spans at least the "missing dimension" from $H^{p-1}$. The theorem of Sard tells me that the critical points of every differentiable map between two differentiable manifolds have measure zero. I don't know which map I have to use. Can someone give me a hint?
HINT: Consider the map $F\colon M\times S^{p-1} \to\Bbb R$ given by $F(x,v) = x\cdot v$. This is a submersion. Therefore, for almost all $v\in S^{p-1}$, $0$ is a regular value of $F_v\colon M\to\Bbb R$.