Let $ U\subseteq \mathbb{R}^n $ open and $ f\in \mathcal{C}^1(U) $ with $ c>0 $ such that for all $ x,y\in U $ it applies $ \|f(x)-f(y)\|\geq c \cdot \|x-y\| $. Now let $ Df(x)\in \text{GL}_n(\mathbb{R}) $ for all $ x\in U $.
If $ U=\mathbb{R}^n $ then $ f $ is bijective.
My idea(s):
1.) To show $ f $ is injective I use: For all $ x,y\in \mathbb{R}^n $ with $ x\neq y $ it applies $ 0<c\cdot \|x-y\|\leq \|f(x)-f(y)\| $ such that $ f(x)\neq f(y) $ and I'm done. But I don't any idea how to show $ f $ surjective.
2.) I try to use the Inverse function theorem. But I have the following problem with that wich I cannot solve. This theorem inculdes that for a point $ x_0\in U $ there exists an open set $ \tilde{U}\subseteq U $ such that $ f_{|\tilde{U}} \tilde{U}\to f(\tilde{U}) $ is bijective and the local inverse of f called $ g:f(\tilde{U})\to \tilde{U} $ is continuously differentiable. But I don't see why it would apply that $ f(\mathbb{R}^n)=\mathbb{R}^n $. Here I just have the input value $ x_0\in \mathbb{R}^n $ which doesn't help me especially to show f is surjective.
