Canonical projection on submanifold $M^k $ over a hyperplane $H^{n}$ is immersion

Question

Let $M^k \subset \mathbb{R}^{n+1}$, $M$ compact and $2k\leq n$. Show tha exist a $n$-hyperplane $H^n\subset \mathbb{R}^{n+1}$ such that if $\pi:H^{n}\oplus (H^n)^{\perp}\rightarrow H^{n}$ is the projection then $\pi\vert_{M}:M\rightarrow H^{n}$ is an immersion.
This is a exercise where suppose to use Sard's Theorem, let me a hint please.

Answer 1

HINT: Can you use Sard's Theorem to show that there is a unit vector in $\Bbb R^{n+1}$ that belongs to no tangent space $T_xM$?