If $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$, then $f$ is a diffeomorphism Suppose that $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ is a differentiable function such that $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$. (Note that $\mathbf{J}_f$ is the Jacobian matrix, $I_n$ is the identity matrix and $||M|| = \big( \sum m_{ij}^2 \big)^{1/2}$). Then I want to prove that $f$ is a diffeomorphism. 
I guess I should apply Implicit function theorem but I do not know how to bring the condition $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ into play. 
 A: According to the Wikipedia-Article (http://en.m.wikipedia.org/wiki/Matrix_norm#Frobenius_norm)
linked in the comments, we have
$$
\left\Vert {\rm id}-Df\left(x\right)\right\Vert _{2}=\left\Vert Df\left(x\right)-{\rm id}\right\Vert _{2}\leq\left\Vert Df\left(x\right)-{\rm id}\right\Vert \leq\frac{1}{2n}<1.
$$
Using a standard Neumann-series argument, this implies that the Jacobian
$$
Df\left(x\right)={\rm id}-\left({\rm id}-Df\left(x\right)\right)
$$
is invertible (with inverse $\left(Df\left(x\right)\right)^{-1}=\sum_{n=0}^{\infty}\left({\rm id}-Df\left(x\right)\right)^{n}$)
for all $x\in\mathbb{R}^{n}$. Hence, the inverse function theorem
implies that $f$ is a diffeomorphism iff it is bijective (because
it is a local diffeomorphism).

To show bijectivity, fix $p\in\mathbb{R}^{n}$ and consider the map
$$
F_{p}:\mathbb{R}^{n}\to\mathbb{R}^{n},x\mapsto x-f\left(x\right)+p.
$$
It suffices to show that $F_{p}$ has a unique fixed point $x_{0}\in\mathbb{R}^{n}$,
because this yields
$$
x_{0}=F_{p}\left(x_{0}\right)=x_{0}-f\left(x_{0}\right)+p\qquad\Longleftrightarrow\qquad f\left(x_{0}\right)=p,
$$
so that $f$ is surjective. By uniqueness of $x_{0}$, we also get
injectivity of $f$.
If you need further hints, see the spoilers below.

 To show that $F_{p}$ has a unique fixed point, use Banach's fixed point theorem (I leave the details to you).

$ $

 To show that $F_p$ is contractive, take the derivative and use the mean value theorem in the multidimensional, vector-valued version (http://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorem_for_vector-valued_functions).

