Let $f:\mathbb{R}^n \longrightarrow \mathbb{R}^n,f\in C^1(\mathbb{R}^n)$ such that $\forall x \in \mathbb{R}^n\;,\;f'(x)$ is an isomorphism and:

$$ \|\left(f'(x)\right)^{-1}\|\le 1\;,\forall x \in \mathbb{R}^n$$

How can we prove that $f$ is a diffeomorphism ? 

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Author's suggestion: 

$\forall [f(a),\color{red}b]\subset \mathbb{R}^n,\exists \lambda:[0,1]\longrightarrow \mathbb{R}^n$ such that $\lambda(0)=a$ and $f(\lambda(t))=(1-t)f(a)+t\color{red}b$

therefore $f$ is surjective.

Note that $f$ is a local diffeomorphism, the strong part is to show that $f$ is injective.

There is a proof without using homotopy ?

Any hints would be appreciated.