Let $S^{n-1}=\{x\in \mathbb R^n:||x||=1\}=\partial B_1(0)$. I have to prove that there exists a Lipschitz function $d: \mathbb R^{n-1} \to S^{n-1}$ i.e. $$\exists L\ \forall x,y \in \mathbb R^{n-1},\ ||d(x)-d(y)||\le L||x-y||$$ My effort:
1) $S^{n-1}$ is a compact set of $\mathbb R^n$ because is colsed and bounded 2) $S^{n-1}=\partial B_1(0)$, which is a Lipschitz domain (well known)
so I thinked to use the compactness to find a finite cover $\mathcal S$ of $S^{n-1}$, such that for every element of $S\in \mathcal S$: $$\exists f\ \ \ A_S=f(S),\ A_S\in \mathbb R^{n-1}$$ with $A_S$ bounded and $f(\cdot)$ Lipschitz. I cannot formalize and finish this proof and I'm wondering if it is the simplest way to proceed. Do you have any other idea? Can you improve this one?