Transversal intersection of one-dimensional submanifold with $S^{n-1}$

Question

I need some help regarding the following exercise

Let $f:\mathbb{R}^n \to S^{n-1}$ be smooth with $f|_{S^{n-1}}=id_{S^{n-1}}$. Show there exist a one-dimensional submanifold $M\subset\mathbb{R}^n$, which intersects $S^{n-1}$ transversally at exactly one point and s.t. $M\cap D^n$ is compact ($D^n=\{x\in\mathbb{R}^n : |x|\leq 1\}$ )

Thanks for your help