Let $f:\mathbb{R}\to\mathbb{R}$ a $C^1$ function, such that $|f'(t)|\le k<1$ for all $t\in\mathbb{R}$. Define $\varphi:\mathbb{R}^2\to\mathbb{R}^2$ by, $\varphi(x,y)=(x+f(y),y+f(x))$. Show that $\varphi$ is a diffeomorphism.
First I showed that $\varphi$ is injective, then I showed that it is a local diffeomorphism by the inverse function theorem.
Now I have to show, that $\varphi$ is surjective, to conclude that it is a global diffeomorphism.
What I thought was to show that $\varphi(\mathbb{R}^2)$ is open and closed, so it would be $\mathbb{R}^2$.
To show that $\varphi(\mathbb{R}^2)$ is open, I would use the conclusions of the inverse function theorem to conclude that $\varphi(\mathbb{R}^2)$ is the union of open sets, to show that it is closed, it would suffice to show that Cauchy's sequences converge to the set.
In this last part, to show that $\varphi$ is surjective, I think that what I thought is a little laborious, I wanted to know if there is a more direct way to conclude the surjective of $\varphi$.
I ended up encountering this question, but everyone use Cauchy's sequence.